Paraffins

From Rosetta Code
Task
Paraffins
You are encouraged to solve this task according to the task description, using any language you may know.

This organic chemistry task is essentially to implement a tree enumeration algorithm.


Task

Enumerate, without repetitions and in order of increasing size, all possible paraffin molecules (also known as alkanes).


Paraffins are built up using only carbon atoms, which has four bonds, and hydrogen, which has one bond.   All bonds for each atom must be used, so it is easiest to think of an alkane as linked carbon atoms forming the "backbone" structure, with adding hydrogen atoms linking the remaining unused bonds.

In a paraffin, one is allowed neither double bonds (two bonds between the same pair of atoms), nor cycles of linked carbons.   So all paraffins with   n   carbon atoms share the empirical formula     CnH2n+2

But for all   n ≥ 4   there are several distinct molecules ("isomers") with the same formula but different structures.

The number of isomers rises rather rapidly when   n   increases.

In counting isomers it should be borne in mind that the four bond positions on a given carbon atom can be freely interchanged and bonds rotated (including 3-D "out of the paper" rotations when it's being observed on a flat diagram),   so rotations or re-orientations of parts of the molecule (without breaking bonds) do not give different isomers.   So what seem at first to be different molecules may in fact turn out to be different orientations of the same molecule.


Example

With   n = 3   there is only one way of linking the carbons despite the different orientations the molecule can be drawn;   and with   n = 4   there are two configurations:

  •   a   straight   chain:     (CH3)(CH2)(CH2)(CH3)
  •   a branched chain:       (CH3)(CH(CH3))(CH3)


Due to bond rotations, it doesn't matter which direction the branch points in.

The phenomenon of "stereo-isomerism" (a molecule being different from its mirror image due to the actual 3-D arrangement of bonds) is ignored for the purpose of this task.

The input is the number   n   of carbon atoms of a molecule (for instance 17).

The output is how many different different paraffins there are with   n   carbon atoms (for instance   24,894   if   n = 17).

The sequence of those results is visible in the OEIS entry:  

  A00602: number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.

The sequence is (the index starts from zero, and represents the number of carbon atoms):

1, 1, 1, 1, 2, 3, 5, 9, 18, 35, 75, 159, 355, 802, 1858, 4347, 10359,
24894, 60523, 148284, 366319, 910726, 2278658, 5731580, 14490245,
36797588, 93839412, 240215803, 617105614, 1590507121, 4111846763,
10660307791, 27711253769, ...


Extra credit

Show the paraffins in some way.

A flat 1D representation, with arrays or lists is enough, for instance:

*Main> all_paraffins 1
                        [CCP H H H H]
*Main> all_paraffins 2
                        [BCP (C H H H) (C H H H)]
*Main> all_paraffins 3
                        [CCP H H (C H H H) (C H H H)]
*Main> all_paraffins 4
                        [BCP (C H H (C H H H)) (C H H (C H H H)),
                         CCP H (C H H H) (C H H H) (C H H H)]
*Main> all_paraffins 5
                        [CCP H H (C H H (C H H H)) (C H H (C H H H)),
                         CCP H (C H H H) (C H H H) (C H H (C H H H)),
                         CCP (C H H H) (C H H H) (C H H H) (C H H H)]
*Main> all_paraffins 6
                        [BCP (C H H (C H H (C H H H))) (C H H (C H H (C H H H))),
                         BCP (C H H (C H H (C H H H))) (C H (C H H H) (C H H H)),
                         BCP (C H (C H H H) (C H H H)) (C H (C H H H) (C H H H)),
                         CCP H (C H H H) (C H H (C H H H)) (C H H (C H H H)),
                         CCP (C H H H) (C H H H) (C H H H) (C H H (C H H H))]

Showing a basic 2D ASCII-art representation of the paraffins is better; for instance (molecule names aren't necessary):

       methane          ethane               propane               isobutane
                     
          H              H   H              H   H   H              H   H   H
          │              │   │              │   │   │              │   │   │
      H ─ C ─ H      H ─ C ─ C ─ H      H ─ C ─ C ─ C ─ H      H ─ C ─ C ─ C ─ H
          │              │   │              │   │   │              │   │   │
          H              H   H              H   H   H              H   │   H
                                                                       │
                                                                   H ─ C ─ H
                                                                       │
                                                                       H
Links
  •   A paper that explains the problem and its solution in a functional language:

http://www.cs.wright.edu/~tkprasad/courses/cs776/paraffins-turner.pdf

  •   A Haskell implementation:

https://github.com/ghc/nofib/blob/master/imaginary/paraffins/Main.hs

  •   A Scheme implementation:

http://www.ccs.neu.edu/home/will/Twobit/src/paraffins.scm

  •   A Fortress implementation:         (this site has been closed)

http://java.net/projects/projectfortress/sources/sources/content/ProjectFortress/demos/turnersParaffins0.fss?rev=3005

11l

Translation of: Nim
V nMax = 250
V nBranches = 4
V rooted = [BigInt(0)] * (nMax + 1)
V unrooted = [BigInt(0)] * (nMax + 1)
rooted[0] = BigInt(1)
rooted[1] = BigInt(1)
unrooted[0] = BigInt(1)
unrooted[1] = BigInt(1)

F choose(m, k)
   I k == 1
      R m
   V result = m
   L(i) 1 .< k
      result = result * (m + i) I/ (i + 1)
   R result

F tree(br, n, l, sum, cnt)
   V s = 0
   L(b) br + 1 .. :nBranches
      s = sum + (b - br) * n
      I s > :nMax {R}

      V c = choose(:rooted[n], b - br) * cnt

      I l * 2 < s {:unrooted[s] += c}
      I b == :nBranches {R}
      :rooted[s] += c
      L(m) (n - 1 .< 0).step(-1)
         tree(b, m, l, s, c)

F bicenter(s)
   I (s [&] 1) == 0
      :unrooted[s] += :rooted[s I/ 2] * (:rooted[s I/ 2] + 1) I/ 2

L(n) 1 .. nMax
   tree(0, n, n, 1, BigInt(1))
   bicenter(n)
   print(n‘: ’unrooted[n])
Output:
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
...
244: 34576004768296889785887066794910718730985852896707505707076422305798138880427343561380451664648670542260
245: 96356944442415066997623733664230869603611716312377642117711752082444867783045964116385053282421589905891
246: 268540209617944059776303921971316267806082307732727328919911735252505071315985779867844056236080213000010
247: 748434113260252449609376828666343341456610378512725586135955779939163320693444008584504390382026391947130
248: 2086006351917005252913566124773054331962205157167696706926185063169623907656246841866717933958839366769700
249: 5814271898167303040368103945830220447130073898083466852225709084407144308593691069932064987528870826155297
250: 16206624309085062837751018464745815688226709117091506494175397665527493805947344857313038875654104100026504

C

Can't show the tree shapes; count only.

#include <stdio.h>

#define MAX_N 33	/* max number of tree nodes */
#define BRANCH 4	/* max number of edges a single node can have */

/* The basic idea: a paraffin molecule can be thought as a simple tree
   with each node being a carbon atom.  Counting molecules is thus the
   problem of counting free (unrooted) trees of given number of nodes.

   An unrooted tree needs to be uniquely represented, so we need a way
   to cannonicalize equivalent free trees.  For that, we need to first
   define the cannonical form of rooted trees.  Since rooted trees can
   be constructed by a root node and up to BRANCH rooted subtrees that
   are arranged in some definite order, we can define it thusly:
     * Given the root of a tree, the weight of each of its branches is
       the number of nodes contained in that branch;
     * A cannonical rooted tree would have its direct subtrees ordered
       in descending order by weight;
     * In case multiple subtrees are the same weight, they are ordered
       by some unstated, but definite, order (this code doesn't really
       care what the ordering is; it only counts the number of choices
       in such a case, not enumerating individual trees.)

   A rooted tree of N nodes can then be constructed by adding smaller,
   cannonical rooted trees to a root node, such that:
     * Each subtree has fewer than BRANCH branches (since it must have
       an empty slot for an edge to connect to the new root);
     * Weight of those subtrees added later are no higher than earlier
       ones;
     * Their weight total N-1.
   A rooted tree so constructed would be itself cannonical.

   For an unrooted tree, we can define the radius of any of its nodes:
   it's the maximum weight of any of the subtrees if this node is used
   as the root.  A node is the center of a tree if it has the smallest
   radius among all the nodes.  A tree can have either one or two such
   centers; if two, they must be adjacent (cf. Knuth, tAoCP 2.3.4.4).

   An important fact is that, a node in a tree is its sole center, IFF
   its radius times 2 is no greater than the sum of the weights of all
   branches (ibid).  While we are making rooted trees, we can add such
   trees encountered to the count of cannonical unrooted trees.

   A bi-centered unrooted tree with N nodes can be made by joining two
   trees, each with N/2 nodes and fewer than BRANCH subtrees, at root.
   The pair must be ordered in aforementioned implicit way so that the
   product is cannonical. */

typedef unsigned long long xint;
#define FMT "llu"

xint rooted[MAX_N] = {1, 1, 0};
xint unrooted[MAX_N] = {1, 1, 0};

/* choose k out of m possible values; chosen values may repeat, but the
   ordering of them does not matter.  It's binomial(m + k - 1, k) */
xint choose(xint m, xint k)
{
	xint i, r;

	if (k == 1) return m;
	for (r = m, i = 1; i < k; i++)
		r = r * (m + i) / (i + 1);
	return r;
}

/* constructing rooted trees of BR branches at root, with at most
   N radius, and SUM nodes in the partial tree already built. It's
   recursive, and CNT and L carry down the number of combinations
   and the tree radius already encountered. */
void tree(xint br, xint n, xint cnt, xint sum, xint l)
{
	xint b, c, m, s;

	for (b = br + 1; b <= BRANCH; b++) {
		s = sum + (b - br) * n;
		if (s >= MAX_N) return;

		/* First B of BR branches are all of weight n; the
		   rest are at most of weight N-1 */
		c = choose(rooted[n], b - br) * cnt;

		/* This partial tree is singly centered as is */
		if (l * 2 < s) unrooted[s] += c;

		/* Trees saturate at root can't be used as building
		   blocks for larger trees, so forget them */
		if (b == BRANCH) return;
		rooted[s] += c;

		/* Build the rest of the branches */
		for (m = n; --m; ) tree(b, m, c, s, l);
	}
}

void bicenter(int s)
{
	if (s & 1) return;

	/* Pick two of the half-size building blocks, allowing
	   repetition. */
	unrooted[s] += rooted[s/2] * (rooted[s/2] + 1) / 2;
}

int main()
{
	xint n;
	for (n = 1; n < MAX_N; n++) {
		tree(0, n, 1, 1, n);
		bicenter(n);
		printf("%"FMT": %"FMT"\n", n, unrooted[n]);
	}

	return 0;
}

Same idea, with GMP, and done somewhat differently:

#include <gmp.h>
#include <stdio.h>
#include <stdlib.h>

#define MAX_BRANCH 4
#define MAX_N 500

mpz_t bcache[MAX_N + 1];
mpz_t ucache[MAX_N + 1];
mpz_t *rcache[MAX_N + 1][MAX_BRANCH + 1];

mpz_t tmp1, tmp2;
void choose(mpz_t r, mpz_t m, int k)
{
	int i;
	mpz_set(r, m);

	mpz_add_ui(tmp1, m, 1);
	for (i = 1; i < k; ) {
		mpz_mul(r, r, tmp1);
		mpz_divexact_ui(r, r, ++i);

		if (i >= k) break;
		mpz_add_ui(tmp1, tmp1, 1);
	}
}

mpz_t rtmp1, rtmp2;
void calc_rooted(mpz_t res, int n, int b, int r)
{
	mpz_set_ui(res, 0);

	if (n == 1 && b == 0 && r == 0) {
		mpz_set_ui(res, 1);
		return;
	} else if (n <= b || n <= r || n == 1 || b == 0 || r == 0)
		return;

	int b1, r1;
	for (b1 = 1; b1 <= b && r * b1 < n; b1++) {
		choose(rtmp1, bcache[r], b1);
		mpz_set_ui(rtmp2, 0);
		for (r1 = 0; r1 < r && r1 + r * b1 < n; r1++)
			mpz_add(rtmp2, rtmp2, rcache[n - r * b1][b - b1][r1]);
		
		mpz_addmul(res, rtmp1, rtmp2);
	}
}

void calc_first_branch(int n)
{
	int b, r;
	mpz_init_set_ui(bcache[n], 0);

	for (b = 0; b < MAX_BRANCH; b++)
		for (r = 0; r < n; r++)
			mpz_add(bcache[n], bcache[n], rcache[n][b][r]);
}

void calc_unrooted(int n)
{
	int b, r;

	for (b = 0; b <= MAX_BRANCH; b++) {
		mpz_t *p = malloc(sizeof(mpz_t) * n);
		rcache[n][b] = p;
		for (r = 0; r < n; r++) {
			mpz_init(p[r]);
			calc_rooted(p[r], n, b, r);
		}
	}

	calc_first_branch(n);

	mpz_init_set_ui(ucache[n], 0);
	for (r = 0; r * 2 < n; r++)
		for (b = 0; b <= MAX_BRANCH; b++)
			mpz_add(ucache[n], ucache[n], rcache[n][b][r]);
	
	if (!(n & 1)) {
		mpz_add_ui(rtmp1, bcache[n/2], 1);
		mpz_mul(rtmp1, rtmp1, bcache[n/2]);
		mpz_divexact_ui(rtmp1, rtmp1, 2);
		mpz_add(ucache[n], ucache[n], rtmp1);
	}
}

void init(void)
{
	mpz_init(tmp1), mpz_init(tmp2);
	mpz_init(rtmp1), mpz_init(rtmp2);
}

int main(void)
{
	int i;

	init();

	for (i = 0; i <= MAX_N; i++) {
		calc_unrooted(i);
		gmp_printf("%d: %Zd\n", i, ucache[i]);
	}

	return 0;
}

C++

#include <cstdint>
#include <iostream>
#include <vector>

const int32_t MAX_TREE_NODES = 52;
const int32_t MAX_BRANCHES = 4;

std::vector<uint64_t> rooted(MAX_TREE_NODES + 1, 0);
std::vector<uint64_t> unrooted(MAX_TREE_NODES + 1,0);
std::vector<uint64_t> count(MAX_BRANCHES, 0);

void tree(const int32_t& branches, const int32_t& radius, const int32_t& combinations,
		  const int32_t& previous_nodes, const uint64_t& branches_count) {
	
	int32_t nodes = previous_nodes;
	for ( int32_t branch = branches + 1; branch <= MAX_BRANCHES; ++branch ) {
		nodes += radius;

		if ( nodes > MAX_TREE_NODES || ( combinations * 2 >= nodes && branch >= MAX_BRANCHES ) ) {
			return;
		}

		if ( branch == branches + 1 ) {
			count[branches] = rooted[radius] * branches_count;
		} else {
			count[branches] *= ( rooted[radius] + branch - branches - 1 );
			count[branches] /= ( branch - branches );
		}

		if ( combinations * 2 < nodes ) {
			unrooted[nodes] += count[branches];
		}

		if ( branch < MAX_BRANCHES ) {
			rooted[nodes] += count[branches];
		}

		for ( int32_t next_radius = radius - 1; next_radius > 0; --next_radius ) {
			tree(branch, next_radius, combinations, nodes, count[branches]);
		}
	}
}

void bicenter(const int32_t& node) {
	if ( ( node & 1 ) == 0 ) {
		const uint64_t temp = ( rooted[node / 2] + 1 ) * rooted[node / 2];
		unrooted[node] += temp / 2;
	}
}

int main() {
	rooted[0] = rooted[1] = 1;
	unrooted[0] = unrooted[1] = 1;

	for ( int32_t node = 1; node <= MAX_TREE_NODES; ++node ) {
		tree(0, node, node, 1, 1);
		bicenter(node);
		std::cout << node << ": " << unrooted[node] << std::endl;
	}
}
Output:
1: 1
2: 1
3: 1
4: 2
5: 3

// elided

48: 156192366474590639
49: 417612400765382272
50: 1117743651746953270
51: 2994664179967370611
52: 8031081780535296591

D

Translation of: Go
import std.stdio, std.bigint;

enum uint nMax = 250;
enum uint nBranches = 4;

__gshared BigInt[nMax + 1] rooted = [1.BigInt, 1.BigInt /*...*/],
                           unrooted = [1.BigInt, 1.BigInt /*...*/];

void tree(in uint br, in uint n, in uint l, in uint inSum,
          in BigInt cnt) nothrow {
    __gshared static BigInt[nBranches] c;

    uint sum = inSum;
    foreach (immutable b; br + 1 .. nBranches + 1) {
        sum += n;
        if (sum > nMax || (l * 2 >= sum && b >= nBranches))
            return;
        if (b == br + 1) {
            c[br] = rooted[n] * cnt;
        } else {
            c[br] *= rooted[n] + b - br - 1;
            c[br] /= b - br;
        }
        if (l * 2 < sum)
            unrooted[sum] += c[br];
        if (b < nBranches)
            rooted[sum] += c[br];
        foreach_reverse (immutable m; 1 .. n)
            tree(b, m, l, sum, c[br]);
    }
}

void bicenter(in uint s) nothrow {
    if ((s & 1) == 0)
        unrooted[s] += rooted[s / 2] * (rooted[s / 2] + 1) / 2;
}

void main() {
    foreach (immutable n; 1 .. nMax + 1) {
        tree(0, n, n, 1, 1.BigInt);
        n.bicenter;
        writeln(n, ": ", unrooted[n]);
    }
}
Output:
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
...
247: 748434113260252449609376828666343341456610378512725586135955779939163320693444008584504390382026391947130
248: 2086006351917005252913566124773054331962205157167696706926185063169623907656246841866717933958839366769700
249: 5814271898167303040368103945830220447130073898083466852225709084407144308593691069932064987528870826155297
250: 16206624309085062837751018464745815688226709117091506494175397665527493805947344857313038875654104100026504

Run-time with nMax = 250 is about 3.6 seconds (about twice the Go entry using go1.2).

FreeBASIC

Translation of: Pascal
Library: GMP
' version 31-12-2016
' compile with: fbc -s console
' uses gmp, translation from pascal

#Include Once "gmp.bi"

Const As Integer max_n = 500, branch = 4

Dim Shared As mpz_ptr rooted(), unrooted(), c()
Dim Shared As mpz_ptr cnt, tmp

Sub tree(br As UInteger, n As UInteger, l As UInteger, sum As UInteger, cnt As mpz_ptr)

  Dim As UInteger b, m

  For b = br +1 To branch
    sum = sum + n
    If sum > max_n Then Return

    ' prevent unneeded long math
    If (l * 2 >= sum) And (b >= branch) Then Return

    If b = (br +1) Then
      mpz_mul(c(br), rooted(n), cnt)
    Else
      mpz_add_ui(tmp, rooted(n), b - br -1)
      mpz_mul(c(br), c(br), tmp)
      mpz_divexact_ui(c(br), c(br), b - br)
    End If

    If l * 2 < sum Then
      mpz_add(unrooted(sum), unrooted(sum), c(br))
    End If
    If b < branch Then
      mpz_add(rooted(sum), rooted(sum), c(br))
      For m = n -1 To 1 Step -1
        tree(b, m, l, sum, c(br))
      Next
    End If
  Next

End Sub

Sub bicenter(s As UInteger)
  If (s And 1) = 1 Then Return
  mpz_add_ui(tmp, rooted(s \ 2), 1)
  mpz_mul(tmp, rooted(s \ 2), tmp)
  mpz_tdiv_q_2exp(tmp, tmp, 1)
  mpz_add(unrooted(s), unrooted(s), tmp)
End Sub

' ------=< MAIN >=------

Dim As UInteger n, sum
Dim As ZString Ptr ans

ReDim rooted(max_n), unrooted(max_n)
For n = 0 To max_n
    rooted(n) = Allocate(Len(__mpz_struct)) : Mpz_init(  rooted(n))
  unrooted(n) = Allocate(Len(__mpz_struct)) : Mpz_init(unrooted(n))
Next
For n = 0 To 1
  mpz_set_ui(  rooted(n), 1)
  mpz_set_ui(unrooted(n), 1)
Next

ReDim c(branch -1)
For n = 0 To branch -1
  c(n) = Allocate(Len(__mpz_struct)) : Mpz_init(c(n))
Next

cnt = Allocate(Len(__mpz_struct)) : Mpz_init_set_ui(cnt, 1)
tmp = Allocate(Len(__mpz_struct)) : Mpz_init(tmp)

sum = 1
For n = 1 To max_n
  tree(0, n, n, sum, cnt)
  bicenter(n)
  'gmp_printf("%d: %Zd"+Chr(13)+Chr(10), n, unrooted(n))
  ans = Mpz_get_str (0, 10, unrooted(n))
  Print Using "###: "; n; : Print *ans
Next

For n = 0 To max_n
  mpz_Clear(  rooted(n))
  mpz_Clear(unrooted(n))
Next

For n = 0 To branch -1
  mpz_clear(c(n))
Next

mpz_clear(cnt)
mpz_clear(tmp)

' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
  1: 1
  2: 1
  3: 1
  4: 2
  5: 3
  6: 5
  7: 9
  8: 18
  9: 35
 10: 75
 11: 159
 12: 355
 13: 802
 14: 1858
 15: 4347
 16: 10359
 17: 24894
 18: 60523
 19: 148284
 20: 366319
 21: 910726
 22: 2278658
 23: 5731580
 24: 14490245
 25: 36797588
 26: 93839412
 27: 240215803
 28: 617105614
 29: 1590507121
 30: 4111846763
 31: 10660307791
 32: 27711253769
 33: 72214088660
 34: 188626236139
 35: 493782952902
 36: 1295297588128
 37: 3404490780161
 38: 8964747474595
 39: 23647478933969
 40: 62481801147341
 41: 165351455535782
 42: 438242894769226
 43: 1163169707886427
 44: 3091461011836856
 45: 8227162372221203
 46: 21921834086683418
 47: 58481806621987010
 48: 156192366474590639
 49: 417612400765382272
 50: 1117743651746953270
 51: 2994664179967370611
 52: 8031081780535296591
 53: 21557771913572630901
 54: 57919180873148437753
 55: 155745431857549699124
 56: 419149571193411829372
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496: 1134831355904717012865001274054429116248775025280754969468716824949667862347555347996290225366000942364709536217262581087333644261895222727600153755613494310524232705702001879535164138079533205048872360533370572443014
497: 3179024139613983878848518783709510209263571759925674510120249336292840845927230735400333746765269611853557998210338782704187073085423815480824362906609395745543530469335794431219957388926521344251394881172155468218720
498: 8905549466780876642396073343654258905917693028466999354614122414334803973053490695667380301725459866905052509157458138657523790806693604983304945043446074074823147488033014740909049433466213254508146498309733349188106
499: 24947785035695184159749164581713850084342937257706468262391560672422449715753806019752051214121544475469837920461991186908900888093113539339801359611535867899901287200789925164011521414091897974716359811774035382416687
500: 69888806977968318652007568872323509883145559195919337637376568556764896550639165374194198834566064897188044072953504164443147167221896769574954040005634802439024034052447587053721967608928388404735140565376686372508743

Go

Translation of: C
package main

import (
    "fmt"
    "math/big"
)

const branches = 4
const nMax = 500

var rooted, unrooted [nMax + 1]big.Int
var c [branches]big.Int
var tmp = new(big.Int)
var one = big.NewInt(1)

func tree(br, n, l, sum int, cnt *big.Int) {
    for b := br + 1; b <= branches; b++ {
        sum += n
        if sum > nMax {
            return
        }
        if l*2 >= sum && b >= branches {
            return
        }
        if b == br+1 {
            c[br].Mul(&rooted[n], cnt)
        } else {
            tmp.Add(&rooted[n], tmp.SetInt64(int64(b-br-1)))
            c[br].Mul(&c[br], tmp)
            c[br].Div(&c[br], tmp.SetInt64(int64(b-br)))
        }
        if l*2 < sum {
            unrooted[sum].Add(&unrooted[sum], &c[br])
        }
        if b < branches {
            rooted[sum].Add(&rooted[sum], &c[br])
        }
        for m := n - 1; m > 0; m-- {
            tree(b, m, l, sum, &c[br])
        }
    }
}

func bicenter(s int) {
    if s&1 == 0 {
        tmp.Rsh(tmp.Mul(&rooted[s/2], tmp.Add(&rooted[s/2], one)), 1)
        unrooted[s].Add(&unrooted[s], tmp)
    }
}

func main() {
    rooted[0].SetInt64(1)
    rooted[1].SetInt64(1)
    unrooted[0].SetInt64(1)
    unrooted[1].SetInt64(1)
    for n := 1; n <= nMax; n++ {
        tree(0, n, n, 1, big.NewInt(1))
        bicenter(n)
        fmt.Printf("%d: %d\n", n, &unrooted[n])
    }
}

Output: (trimmed)

1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
22: 2278658
23: 5731580
24: 14490245
25: 36797588
26: 93839412
27: 240215803
28: 617105614
29: 1590507121
30: 4111846763
31: 10660307791
32: 27711253769
33: 72214088660
34: 188626236139
35: 493782952902
...
499: 2494778503569518415974916458171385008434293725770646826239156067
242244971575380601975205121412154447546983792046199118690890088809311
353933980135961153586789990128720078992516401152141409189797471635981
1774035382416687
500: 6988880697796831865200756887232350988314555919591933763737656855
676489655063916537419419883456606489718804407295350416444314716722189
676957495404000563480243902403405244758705372196760892838840473514056
5376686372508743

Haskell

Using formula from OEIS page, similar to the Mathematica entry below:

-- polynomial utils
a `nmul` n = map (*n) a
a `ndiv` n = map (`div` n) a

instance (Integral a) => Num [a] where
  (+) = zipWith (+)
  negate = map negate
  a * b = foldr f undefined b where
    f x z = (a `nmul` x) + (0 : z)
  abs _ = undefined
  signum _ = undefined
  fromInteger n = fromInteger n : repeat 0

-- replace x in polynomial with x^n
repl a n = concatMap (: replicate (n-1) 0) a

-- S2: (a^2 + b)/2
cycleIndexS2 a b = (a*a + b)`ndiv` 2

-- S4: (a^4 + 6 a^2 b + 8 a c + 3 b^2 + 6 d) / 24
cycleIndexS4 a b c d =	((a ^ 4) +
			 (a ^ 2 * b) `nmul` 6 +
			 (a * c) `nmul` 8 +
			 (b ^ 2) `nmul` 3 +
			 d `nmul` 6) `ndiv` 24


a598 = x1
-- A000598: A(x) = 1 + (1/6)*x*(A(x)^3 + 3*A(x)*A(x^2) + 2*A(x^3))
x1 = 1 : ((x1^3) + ((x2*x1)`nmul` 3) + (x3`nmul`2)) `ndiv` 6
x2 = x1`repl`2
x3 = x1`repl`3
x4 = x1`repl`4

-- A000678 = x CycleIndex(S4, A000598(x))
a678 = 0 : cycleIndexS4 x1 x2 x3 x4

-- A000599 = CycleIndex(S2, A000598(x) - 1)
a599 = cycleIndexS2 (0 : tail x1) (0 : tail x2)

-- A000602 = A000678(x) - A000599(x) + A000599(x^2)
a602 = a678 - a599 + x2

main = mapM_ print $ take 200 $ zip [0 ..] a602

Counting trees with some fairly primitive caching:

import Data.Array

choose :: Integer -> Int -> Integer
choose m k = let kk = toInteger k in (product [m..m+kk-1]) `div` (product [1..kk])

max_branches = 4
max_nodes = 200

bcache = listArray (0, max_nodes)
	[sum[rcache!n!b!r | r <- [0..n], b <- [0..max_branches-1]] | n <- [0..max_nodes]]
build_block = (bcache !)

rcache = listArray (0,max_nodes) [arr_b i | i <- [0..max_nodes]] where
	arr_b n = listArray(0,max_branches) [arr_r b n | b <- [0..max_branches]]
	arr_r b n = listArray(0,n) [rooted n b r | r <- [0..n]]

rooted 1 0 0 = 1
rooted 1 _ _ = 0
rooted _ 0 _ = 0
rooted _ _ 0 = 0
rooted n b r
	| (n <= b) || (n <= r) = 0
	| otherwise = sum [(firsts b1) * (rests b1) | b1 <- [1..b], r * b1 < n] where
		firsts = choose (build_block r)
		rests bb = sum [rcache!(n-r*bb)!(b - bb)!r1 | r1 <- [0..r-1], r1 < (n-r*bb)]

unrooted n = unicenter + bycenter where
	unicenter = sum [ rcache!n!b!r | b <- [0..max_branches], r <-[0..n], r * 2 < n]
	bycenter| odd n = 0
		| otherwise = x * (x + 1) `div` 2 where x = build_block (n `div` 2)

main = mapM_ print $ map (\x->(x, unrooted x)) [1..max_nodes]
Output:
(1,1)
(2,1)
(3,1)
(4,2)
(5,3)
(6,5)
(7,9)
(8,18)
(9,35)
(10,75)
(11,159)
(12,355)
(13,802)
(14,1858)
...
(199,339176261988518728096836182493660862745709169352281541101577697702699073887422989905)
(200,943043328799038505167332910595466006794464252841664581909549826351576307818857723954)

J

The following code is an interpretation of the Haskell program listed in the links above.

part3=: ;@((<@([(],.(-+/"1))],.]+i.@(]-~1+<.@-:@-))"0 i.@>:@<.@%&3))

part4=: 3 :0
ij=.; (,.]+i.@:(]-~1+[:<.3%~y-]))&.> i.1+<.y%4
(,.y - +/"1) ; (<@(],"1 0 <.@-:@(y-[) (] + i.@>:@-) {:@] >. (>.-:y)-[)~+/)"1 ij 
)

c0=: */@:{
c1=: 13 :'(*-:@(*>:))/y{~}:x'
c2=: 13 :'(*-:@(*>:))~/y{~}.x'
c3=: 13 :'3!2+y{~{.x'

radGenN=: [:;[:(],[:+/c0`c1`c2`c3@.(#.@(}.=}:)@[)"1)&.>/(<1x),~part3&.>@ i.@-

bcpGenN=: [: , 0 ,.~ -:@(*>:)@({~i.)

c11=: 13 :'*/(y{~0 1{x), -:(*>:)y{~{:x'
c12=: 13 :'*/(y{~0 3{x), -:(*>:)y{~2{x'
c13=: 13 :'*/(y{~{.x) , 3!2+ y{~{: x'
c14=: 13 :'*/(y{~_2{.x), -:(*>:)y{~{.x'
c15=: 13 :'*/ -:(*>:) y{~0 3{x'
c16=: 13 :'*/(y{~{:x) , 3!2+ y{~{. x'
c17=: 13 :'4!3+y{~{.x'

cassl=: c0`c11`c12`c13`c14`c15`c16`c17

ccpGenN=: 4 :0
if. 0=y do. i.0 return. end.
y{.2({.,0,}.) 0,+/@:(x cassl@.(#.@(}.=}:)@[)"1~[)@:part4"0 [1-.~i.y-1
)

NofParaff=: {. radGenN ((ccpGenN +:) + bcpGenN ) 2&|+<.@-:

Output:

   6 6 $ NofParaff 36
         1           1           1           1            2            3
         5           9          18          35           75          159
       355         802        1858        4347        10359        24894
     60523      148284      366319      910726      2278658      5731580
  14490245    36797588    93839412   240215803    617105614   1590507121
4111846763 10660307791 27711253769 72214088660 188626236139 493782952902

Java

Translation of C via Go and D

import java.math.BigInteger;
import java.util.Arrays;

class Test {
    final static int nMax = 250;
    final static int nBranches = 4;

    static BigInteger[] rooted = new BigInteger[nMax + 1];
    static BigInteger[] unrooted = new BigInteger[nMax + 1];
    static BigInteger[] c = new BigInteger[nBranches];

    static void tree(int br, int n, int l, int inSum, BigInteger cnt) {
        int sum = inSum;
        for (int b = br + 1; b <= nBranches; b++) {
            sum += n;

            if (sum > nMax || (l * 2 >= sum && b >= nBranches))
                return;

            BigInteger tmp = rooted[n];
            if (b == br + 1) {
                c[br] = tmp.multiply(cnt);
            } else {
                c[br] = c[br].multiply(tmp.add(BigInteger.valueOf(b - br - 1)));
                c[br] = c[br].divide(BigInteger.valueOf(b - br));
            }

            if (l * 2 < sum)
                unrooted[sum] = unrooted[sum].add(c[br]);

            if (b < nBranches)
                rooted[sum] = rooted[sum].add(c[br]);

            for (int m = n - 1; m > 0; m--)
                tree(b, m, l, sum, c[br]);
        }
    }

    static void bicenter(int s) {
        if ((s & 1) == 0) {
            BigInteger tmp = rooted[s / 2];
            tmp = tmp.add(BigInteger.ONE).multiply(rooted[s / 2]);
            unrooted[s] = unrooted[s].add(tmp.shiftRight(1));
        }
    }

    public static void main(String[] args) {
        Arrays.fill(rooted, BigInteger.ZERO);
        Arrays.fill(unrooted, BigInteger.ZERO);
        rooted[0] = rooted[1] = BigInteger.ONE;
        unrooted[0] = unrooted[1] = BigInteger.ONE;

        for (int n = 1; n <= nMax; n++) {
            tree(0, n, n, 1, BigInteger.ONE);
            bicenter(n);
            System.out.printf("%d: %s%n", n, unrooted[n]);
        }
    }
}
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
(...)
248: 2086006351917005252913566124773054331962205157167696706926185063169623907656246841866717933958839366769700
249: 5814271898167303040368103945830220447130073898083466852225709084407144308593691069932064987528870826155297
250: 16206624309085062837751018464745815688226709117091506494175397665527493805947344857313038875654104100026504

jq

Works with: jq version 1.4

The following version is based on the C/python/ruby implementations.

Currently jq uses IEEE 754 64-bit numbers. Large integers are approximated by floats and therefore the results generated by the program presented here are only precise for n up to and including 45.

def MAX_N: 500;  # imprecision begins at 46
def BRANCH: 4;

# state: [unrooted, ra]
# tree(br; n; l; sum; cnt) where initially: l=n, sum=1 and cnt=1
def tree(br; n; l; sum; cnt):

  # The inner function is used to implement the range(b+1; BRANCH) loop
  # as there are early exits.
  # On completion, _tree returns [unrooted, ra]
  def _tree: # state [ (b, c, sum),  (unrooted, ra)]
    if length != 5 then error("_tree input has length \(length)") else . end
    | .[0] as $b | .[1] as $c | .[2] as $sum | .[3] as $unrooted | .[4] as $ra
    | if $b > BRANCH then [$unrooted, $ra]
      else
        ($sum + n) as $sum
        | if $sum >= MAX_N or
             # prevent unneeded long math
             ( l * 2 >= $sum and $b >= BRANCH) then [$unrooted, $ra]                      # return
          else (if $b == br + 1 then $ra[n] * cnt 
                else ($c * ($ra[n] + (($b - br - 1)))) / ($b - br) | floor
                end) as $c
          | (if l * 2 < $sum then ($unrooted | .[$sum] += $c)
             else $unrooted end) as $unrooted
          | if $b >= BRANCH then [$b+1, $c, $sum, $unrooted, $ra] | _tree                 # next
            else  [$unrooted, ($ra | .[$sum] += $c) ]
            | reduce range(1; n) as $m (.;  tree($b; $m; l; $sum; $c)) 
            | ([$b + 1, $c, $sum] + .) | _tree 
            end
          end
      end
  ;

  # start by incrementing b, and prepending values for (b,c,sum)
  ([br+1, cnt, sum] + .)  | _tree
;
 
# input and output: [unrooted, ra]
def bicenter(s):
  if s % 2 == 1 then .
  else
     .[1][s / 2] as $aux
     | .[0][s] += ($aux * ($aux + 1)) / 2 # 2 divides odd*even
  end
;
 
def array(n;init): [][n-1] = init | map(init);

def ra: array( MAX_N; 0) | .[0] = 1 | .[1] = 1;

def unrooted: ra;

# See below for a simpler implementation using "foreach"
def paraffins:
  # range(1; MAX_N)
  def _paraffins(n):
    if n >= MAX_N then empty
    else tree(0; n; n; 1; 1) | bicenter(n)
    | [n, .[0][n]],  # output
      _paraffins(n+1)
    end;
  [unrooted, ra] | _paraffins(1)
;

paraffins

Output (trimmed):

$ jq -M -n -c -f paraffins.jq
[1,1]
[2,1]
[3,1]
[4,2]
[5,3]
[6,5]
[7,9]
[8,18]
[9,35]
[10,75]
[11,159]
[12,355]
[13,802]
[14,1858]
[15,4347]
[16,10359]
[17,24894]
[18,60523]
[19,148284]
[20,366319]
[21,910726]
[22,2278658]
[23,5731580]
[24,14490245]
[25,36797588]
[26,93839412]
[27,240215803]
[28,617105614]
[29,1590507121]
[30,4111846763]
[31,10660307791]
[32,27711253769]
[33,72214088660]
[34,188626236139]
[35,493782952902]
[36,1295297588128]
[37,3404490780161]
[38,8964747474595]
[39,23647478933969]
[40,62481801147341]
[41,165351455535782]
[42,438242894769226]
[43,1163169707886427]
[44,3091461011836856]
[45,8227162372221203]
# The above answer for 45 is the last precisely correct answer -- floating point approximations hereon:
...
[100,5.921072038125814e+39]
...
[499,2.4947785035695037e+217]

Using foreach

Works with: jq version >1.4

The following is a more elegant alternative to paraffins/0 as defined above but requires "foreach":

def paraffins:
  foreach range(1; MAX_N) as $n
    ( [unrooted, ra];
      tree(0; $n; $n; 1; 1) | bicenter($n);
      [$n, .[0][$n]]
    )
;

Julia

Translation of: Go

Output is the same as the Go version.

const branches = 4
const nmax = 500

const rooted = zeros(BigInt, nmax + 1)
const unrooted = zeros(BigInt, nmax + 1)
rooted[1] = rooted[2] = unrooted[1] = unrooted[2] = 1
const c = zeros(BigInt, branches)

function tree(br, n, l, sum, cnt)
    for b in br+1:branches
        sum += n
        if (sum > nmax) || (l * 2 >= sum && b >= branches)
            return
        elseif b == br + 1
            c[br + 1] = rooted[n + 1] * cnt
        else
            c[br + 1] *= rooted[n + 1] + b - br - 1
            c[br + 1] = div(c[br + 1], b - br)
        end
        if l*2 < sum
            unrooted[sum + 1] += c[br + 1]
        end
        if b < branches
            rooted[sum + 1] += c[br + 1]
        end
        for m in n-1:-1:1
            tree(b, m, l, sum, c[br + 1])
        end
    end
end

bicenter(n) = if iseven(n) unrooted[n + 1] += div(rooted[div(n, 2) + 1] * (rooted[div(n, 2) + 1] + 1), 2) end

function paraffins()
    for n in 1:nmax
        tree(0, n, n, 1, one(BigInt))
        bicenter(n)
        println("$n: $(unrooted[n + 1])")
    end
end

paraffins()

Kotlin

Translation of: Java
// version 1.1.4-3

import java.math.BigInteger

const val MAX_N = 250
const val BRANCHES = 4

val rooted   = Array(MAX_N + 1) { if (it < 2) BigInteger.ONE else BigInteger.ZERO }
val unrooted = Array(MAX_N + 1) { if (it < 2) BigInteger.ONE else BigInteger.ZERO }
val c = Array(BRANCHES) { BigInteger.ZERO }

fun tree(br: Int, n: Int, l: Int, s: Int, cnt: BigInteger) {
    var sum = s
    for (b in (br + 1)..BRANCHES) {
        sum += n
        if (sum > MAX_N || (l * 2 >= sum && b >= BRANCHES)) return

        var tmp = rooted[n]
        if (b == br + 1) {
            c[br] = tmp * cnt
        }
        else {
            val diff = (b - br).toLong()
            c[br] *= tmp + BigInteger.valueOf(diff - 1L)
            c[br] /= BigInteger.valueOf(diff)
        }
 
        if (l * 2 < sum) unrooted[sum] += c[br]
        if (b < BRANCHES) rooted[sum] += c[br]
        for (m in n - 1 downTo 1) tree(b, m, l, sum, c[br])
    }
}

fun bicenter(s: Int) {
    if ((s and 1) == 0) {
        var tmp = rooted[s / 2]
        tmp *= tmp + BigInteger.ONE
        unrooted[s] += tmp.shiftRight(1)
    }
}

fun main(args: Array<String>) {
    for (n in 1..MAX_N) {
        tree(0, n, n, 1, BigInteger.ONE)
        bicenter(n)
        println("$n: ${unrooted[n]}")
    }
}
Output:
Same as Java entry

Mathematica /Wolfram Language

Works with: Mathematica version 9.0

Using the formula on OEIS.

s[m_, p_, n_] := 
  CycleIndexPolynomial[SymmetricGroup[m], 
   Table[ComposeSeries[p, x^i + O[x]^(n + 1)], {i, m}]];
G000598[n_] := Nest[1 + x s[3, #, n] &, 1 + O[x], n];
G000602[n_] := 
  x s[4, #, n] - s[2, # - 1, n] + 
     ComposeSeries[#, x^2 + O[x]^(n + 1)] &[G000598[n]];
A000602[n_] := SeriesCoefficient[G000602[n], n];
A000602List[n_] := CoefficientList[G000602[n], x];
Grid@Transpose@{Range[0, 200], A000602List@200}
Output:
0	1
1	1
2	1
3	1
4	2
5	3
6	5
7	9
8	18
9	35
10	75
11	159
12	355
13	802
...
199	339176261988518728096836182493660862745709169352281541101577697702699073887422989905
200	943043328799038505167332910595466006794464252841664581909549826351576307818857723954

Nim

Translation of: C
Library: bigints
import bigints

const
  nMax: int32 = 250
  nBranches = 4

var rooted, unrooted: array[nMax + 1, BigInt]
rooted[0..1] = [1.initBigInt, 1.initBigInt]
unrooted[0..1] = [1.initBigInt, 1.initBigInt]
for i in 2 .. nMax:
  rooted[i] = 0.initBigInt
  unrooted[i] = 0.initBigInt

proc choose(m: BigInt; k: int32): BigInt =
  result = m
  if k == 1: return
  for i in 1 ..< k:
    result = result * (m + i) div (i + 1)

proc tree(br, n, l, sum: int32; cnt: BigInt) =
  var s: int32 = 0
  for b in br + 1 .. nBranches:
    s = sum + (b - br) * n
    if s > nMax: return

    let c = choose(rooted[n], b - br) * cnt

    if l * 2 < s: unrooted[s] += c
    if b == nBranches: return
    rooted[s] += c
    for m in countdown(n-1, 1):
      tree b, m, l, s, c

proc bicenter(s: int32) =
  if (s and 1) == 0:
    unrooted[s] += rooted[s div 2] * (rooted[s div 2] + 1) div 2

for n in 1 .. nMax:
  tree 0, n, n, 1, 1.initBigInt
  n.bicenter
  echo n, ": ", unrooted[n]
Output:
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
...
249: 5814271898167303040368103945830220447130073898083466852225709084407144308593691069932064987528870826155297
250: 16206624309085062837751018464745815688226709117091506494175397665527493805947344857313038875654104100026504

PARI/GP

This function is for recent PARI/GP:

paraffin(p) =
{
  local (P = p+1, R, U = R = Vec([1,1], P));

  for (n = 1, p,
    ((B,n,C,S,l=n) -> my(b,c,i,s);
      for (b = 1, 4-B,
        if ((s = S + b * n) < P,
          c = R[n+1] * C * prod(i = 1, b-1, (R[n+1]+i)/(i+1));
          if (l+l < s, U[s+1] += c);
          if (B+b < 4, R[s+1] += c; i = n; while (i--, self()(B+b, i, c, s, l)))))
    )(0,n,1,1);
    if (n % 2,, U[n+1] += R[n/2+1] * (R[n/2+1]+1)/2);
    print([n, U[n+1]]))
}


Code for older version of PARI/GP < 2.9:

iso(B,n,C,S,l=n) =
{
  my (b,c,i,s);

  for (b = 1, 4-B,
    if ((s = S + b * n) < P,
      c = R[n+1] * C * prod(i = 1, b-1, (R[n+1]+i)/(i+1));
      if (l+l < s, U[s+1] += c);
      if (B+b < 4, R[s+1] += c; i = n; while (i--, iso(B+b, i, c, s, l)))))
}

paraffin(p) =
{
  local (P = p+1, R, U = R = Vec([1,1], P));

  for (n = 1, p, iso(0, n, 1, 1);
    if (n % 2,, U[n+1] += R[n/2+1] * (R[n/2+1]+1)/2);
    print([n, U[n+1]]))
}

Output:

paraffin(32)

[1, 1]
[2, 1]
[3, 1]
[4, 2]
[5, 3]
[6, 5]
[7, 9]
[8, 18]
[9, 35]
[10, 75]
[11, 159]
[12, 355]
[13, 802]
[14, 1858]
[15, 4347]
[16, 10359]
[17, 24894]
[18, 60523]
[19, 148284]
[20, 366319]
[21, 910726]
[22, 2278658]
[23, 5731580]
[24, 14490245]
[25, 36797588]
[26, 93839412]
[27, 240215803]
[28, 617105614]
[29, 1590507121]
[30, 4111846763]
[31, 10660307791]
[32, 27711253769]

Pascal

Works with: Free_Pascal
Library: GMP

Conversion of the C example:

Program Paraffins;

uses
  gmp;
  
const
  max_n = 500;
  branch = 4;

var
  rooted, unrooted: array [0 .. max_n-1] of mpz_t;
  c: array [0 .. branch-1] of mpz_t;
  cnt, tmp: mpz_t;
  n: integer;
  fmt: pchar;
  sum: integer;

procedure tree(br, n, l: integer; sum: integer; cnt: mpz_t);
  var
    b, m: integer;
  begin
    for b := br + 1 to branch do
    begin
      sum := sum + n;
      if sum >= max_n then
	exit;

      (* prevent unneeded long math *)
      if (l * 2 >= sum) and (b >= branch) then 
	exit;

      if b = (br + 1) then
	mpz_mul(c[br], rooted[n], cnt)
      else
      begin
	mpz_add_ui(tmp, rooted[n], b - br - 1);
	mpz_mul(c[br], c[br], tmp);
	mpz_divexact_ui(c[br], c[br], b - br);
      end;

      if l * 2 < sum then
	mpz_add(unrooted[sum], unrooted[sum], c[br]);

      if b < branch then
      begin
	mpz_add(rooted[sum], rooted[sum], c[br]);
	for m := n-1 downto 1 do
	  tree(b, m, l, sum, c[br]);
      end;
    end;
  end;

procedure bicenter(s: integer);
begin
  if odd(s) then
    exit;
  mpz_add_ui(tmp, rooted[s div 2], 1);
  mpz_mul(tmp, rooted[s div 2], tmp);
  mpz_tdiv_q_2exp(tmp, tmp, 1);

  mpz_add(unrooted[s], unrooted[s], tmp);
end;

begin
  for n := 0 to 1 do
  begin
    mpz_init_set_ui(rooted[n], 1);
    mpz_init_set_ui(unrooted[n], 1);
  end;
  for n := 2 to max_n-1 do
  begin
    mpz_init_set_ui(rooted[n], 0);
    mpz_init_set_ui(unrooted[n], 0);
  end;
  for n := 0 to BRANCH-1 do
    mpz_init(c[n]);

  mpz_init(tmp);

  mpz_init_set_ui(cnt, 1);
  sum := 1;
  for n := 1 to MAX_N do
  begin
    tree(0, n, n, sum, cnt);
    bicenter(n);
    mp_printf('%d: %Zd'+chr(13)+chr(10), n, @unrooted[n]);
  end;
end.

Output (trimmed):

1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
22: 2278658
23: 5731580
24: 14490245
25: 36797588
26: 93839412
27: 240215803
28: 617105614
29: 1590507121
30: 4111846763
31: 10660307791
32: 27711253769
33: 72214088660
34: 188626236139
35: 493782952902
..
499: 2494778503569518415974916458171385008434293725770646826239156067242244971575380601975205121412154447546983792
     0461991186908900888093113539339801359611535867899901287200789925164011521414091897974716359811774035382416687
500: 35027241765694350953134643689731689510165490186734465975716452976264824623941928748229034890456068034873862265
     02245470029705798383457667757879016306516643706616463708365586217812176623950823312143595975577679489832553344

Alternative method

Works with: Free_Pascal

This method of counting alkanes is based on a paper by Shinsaku Fujita (see program for reference). Multi-precision is not used, so alkanes are counted only up to order 49. The results are identical to those from the REXX program.

program CountAlkanes;

{$mode objfpc}{$H+}

uses SysUtils; // only for output

type TArrayUint64 = array of uint64;
{
  Function to count alkanes, based on: Shinsaku Fujita,
  "Numbers of Alkanes and Monosubstituted Alkanes.
  A Long-Standing Interdisciplinary Problem over 130 Years",
  Bull. Chem. Soc. Jpn. Vol. 83, No. 1, 1–18 (2010)
}
function CountAlkanes() : TArrayUint64;
const
  MAX_RESULT_INDEX = 49; // as far as this code can get without multi-precision
  MAX_R_INDEX = MAX_RESULT_INDEX div 2;
var
  R : array [0..MAX_R_INDEX] of uint64;
  nrCentUnb : uint64; // number of centroidal unbalanced alkanes
  temp : uint64;
  m, n, h, i, j, k : integer;
begin
  SetLength( result, MAX_RESULT_INDEX + 1); // zero-based
{
  Calculate enough of the coefficients R[], where the generating function
     r(x) = R[0] + R[1]x + R[2]x^2 + R[3]x^3 + ...  satifies
     r(x) = 1 + (x/6)[r(x)^3 + 2r(x^3) + 3r(x)r(x^2)]  (Fujita, equation 4)
}
  R[0] := 1;
  n := 0;
  repeat
    if (n mod 3 = 0) then temp := 2*R[n div 3]
                     else temp := 0;
    for j := 0 to (n div 2) do begin
      inc( temp, 3 * R[j] * R[n - 2*j]);
    end;
    for j := 0 to n do begin
      for k := 0 to (n - j) do begin
        inc(temp, R[j] * R[k] * R[n - j - k]);
      end;
    end;
    Assert( temp mod 6 = 0);  // keep an eye on it
    inc(n);
    R[n] := temp div 6;
  until (n = MAX_R_INDEX);
{
  Now use the generating function
    (x/24)[r(x)^4 + 3r(x^2)^2 + *r(x)r(x^3) + 6r(x)^2r(x^2) + 6r(x^4)]
  where inserting r(x) up to the term in x^m will give the number of alkanes
  of orders 2m+1 and 2m+2, as the coefficients of x^(2m+1) and x^(2m+2).

  Note: In Fujita's paper, equation 23, the factor is 1/24 not x/24,
        but x/24 seems to be needed to give correct results.
}
  result[0] := 1;  // conventional
  for n := 1 to MAX_RESULT_INDEX do begin
    m := (n - 1) div 2; // so n = 2*m + 1 or 2*m + 2
    temp := 0;

    // These loops are written for clarity not efficiency
    for k := 0 to m do begin
      for j := 0 to m do begin
        for i := 0 to m do begin
          h := n - 1 - i - j - k;
          if  (h >= 0) and (h <= m) then inc( temp, R[h]*R[i]*R[j]*R[k]);
        end;
      end;
    end;

    if Odd(n) then begin
      for k := 0 to m do begin
        inc( temp, 3*R[k]*R[m - k]);
      end;
    end;

    for k := 0 to (n - 1) div 3 do begin
      j := n - 1 - 3*k;
      if (j <= m) then inc( temp, 8*R[j]*R[k]);
    end;

    for k := 0 to m do begin
      for j := 0 to m do begin
        i := n - 1 - 2*k - j;
        if (i >= 0) and (i <= m) then inc( temp, 6*R[i]*R[j]*R[k]);
      end;
    end;

    if (n mod 4 = 1) then inc( temp, 6*R[(n - 1) div 4]);

    Assert( temp mod 24 = 0);  // keep an eye on it
    nrCentUnb := temp div 24;
    if Odd(n) then
      result[n] := nrCentUnb
    else begin
      temp := R[n div 2];
      result[n] := nrCentUnb + (temp*(temp + 1) div 2);
    end;
  end;
end;

// Call function and display the results
var
  nrAlkanes : TArrayUint64;
  k : integer;
begin
  nrAlkanes := CountAlkanes();
  for k := 0 to Length( nrAlkanes) - 1 do
    WriteLn( SysUtils.Format( '%2d %d', [k, nrAlkanes[k]]));
end.
Output:
 0 1
 1 1
 2 1
 3 1
 4 2
 5 3
 6 5
[...]
48 156192366474590639
49 417612400765382272

Perl

Translation of: C

This is using Math::GMPz for best performance. Math::GMP works almost as well. Math::BigInt is in core and only 9 times slower for this task.

use Math::GMPz;

my $nmax = 250;
my $nbranches = 4;

my @rooted   = map { Math::GMPz->new($_) } 1,1,(0) x $nmax;
my @unrooted = map { Math::GMPz->new($_) } 1,1,(0) x $nmax;
my @c        = map { Math::GMPz->new(0) } 0 .. $nbranches-1;

sub tree {
  my($br, $n, $l, $sum, $cnt) = @_;
  for my $b ($br+1 .. $nbranches) {
    $sum += $n;
    return if $sum > $nmax || ($l*2 >= $sum && $b >= $nbranches);
    if ($b == $br+1) {
      $c[$br] = $rooted[$n] * $cnt;
    } else {
      $c[$br] *= $rooted[$n] + $b - $br - 1;
      $c[$br] /= $b - $br;
    }
    $unrooted[$sum] += $c[$br] if $l*2 < $sum;
    return if $b >= $nbranches;
    $rooted[$sum] += $c[$br];
    for my $m (reverse 1 .. $n-1) {
      next if $sum+$m > $nmax;
      tree($b, $m, $l, $sum, $c[$br]);
    }
  }
}

sub bicenter {
  my $s = shift;
  $unrooted[$s] += $rooted[$s/2] * ($rooted[$s/2]+1) / 2  unless $s & 1;
}

for my $n (1 .. $nmax) {
  tree(0, $n, $n, 1, Math::GMPz->new(1));
  bicenter($n);
  print "$n: $unrooted[$n]\n";
}

Output identical to C GMP example (truncated to 250).

Phix

Translation of: Ruby
Translation of: FreeBASIC

Using IEEE754 floats, hence imprecise above n=45 on 32 bit, n=52 on 64bit, tested to n=600, giving 3.99378e+262 using %g format, ie accurate to better than 1e-10%.

with javascript_semantics
constant MAX_N = 32,
         BRANCH = 4

sequence rooted = repeat(0,MAX_N+2),
       unrooted = repeat(0,MAX_N+2)

procedure tree(integer br, n, l=n, tot=1, atom cnt=1)
    atom c
    for b=br+1 to BRANCH do
        tot += n
        if tot>=MAX_N+1
        or (l*2>=tot and b>=BRANCH) then
            return
        end if
        integer n1 = n+1,
                t1 = tot+1
        if b==br+1 then
            c = rooted[n1]*cnt
        else
            c *= (rooted[n1]+(b-br-1))/(b-br)
        end if
        if l*2<tot then
            unrooted[t1] += c
        end if
        if b<BRANCH then
            rooted[t1] += c
            for m=1 to n-1 do
                tree(b,m,l,tot,c)
            end for
        end if
    end for
end procedure

procedure bicenter(integer s)
    if even(s) then
        atom aux = rooted[s/2+1]
        s += 1
        unrooted[s] += aux*(aux+1)/2
    end if
end procedure

rooted[1..2] = 1
unrooted[1..2] = 1
for n=1 to MAX_N do
    tree(0, n)
    bicenter(n)
    if n<10 or n=MAX_N then
        printf(1,"%d: %d\n",{n, unrooted[n+1]})
    end if
end for
Output:
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
32: 27711253769

And the same thing using gmp, obviously without any such accuracy limits

with javascript_semantics
constant max_n = 200, branch = 4
atom t0 = time()
 
include mpfr.e
sequence ivals = repeat(1,2)&repeat(0,max_n-1),
        rooted = mpz_inits(max_n+1,ivals),
      unrooted = mpz_inits(max_n+1,ivals),
             c = mpz_inits(branch,0)
mpz tmp = mpz_init()
 
procedure tree(integer br, n, l, s, mpz cnt)
    mpz cbr = c[br]
    for b=br to branch do
        s += n
        if s>max_n 
        or (l*2>=s and b>=branch) then
            return
        end if
        if b=br then
            mpz_mul(cbr, rooted[n+1], cnt)
        else
            mpz_add_ui(tmp, rooted[n+1], b-br)
            mpz_mul(cbr, cbr, tmp)
            mpz_divexact_ui(cbr, cbr, b-br+1)
        end if
        if l*2<s then
            mpz u = unrooted[s+1]
            mpz_add(u, u, cbr)
        end if
        if b<branch then
            mpz r = rooted[s+1]
            mpz_add(r, r, cbr)
            for m=n-1 to 1 by -1 do
                tree(b+1, m, l, s, cbr)
            end for
        end if
    end for
end procedure
 
procedure bicenter(integer s)
    if even(s) then
        mpz aux = rooted[s/2+1],
            u = unrooted[s+1]
        mpz_add_ui(tmp, aux, 1)
        mpz_mul(tmp, aux, tmp)
        mpz_tdiv_q_2exp(tmp, tmp, 1)
        mpz_add(u, u, tmp)
    end if
end procedure
 
mpz cnt = mpz_init(1)
 
integer s = 1
for n=1 to max_n do
    tree(1, n, n, s, cnt)
    bicenter(n)
    if n<=25 or remainder(n,100)=0 then
        printf(1,"%d: %s\n",{n, mpz_get_short_str(unrooted[n+1])})
    end if
end for
?elapsed(time()-t0)

Unusually this is significantly faster (6*) under pwa/p2js than desktop/Phix, the following output/time is from the former.
The limit of 200 in the code above finishes on desktop/Phix in 6s, p2js 1s, 500 takes about 40s in the browser but a whopping three and a half minutes on desktop/Phix, compared to the Go entry above completing that task in 15.6s, all on the same box of course. I have earmarked this task for further investigation into performance improvements, should Phix 2.0 ever actually get started on, that is.

Output:
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
22: 2278658
23: 5731580
24: 14490245
25: 36797588
100: 5921072038125809849884993369103538010139
200: 94304332879903850516...51576307818857723954 (84 digits)
300: 30845274893235665913...77084032710062170279 (129 digits)
400: 13544322063698139999...74867143490557738328 (174 digits)
500: 69888806977968318652...40565376686372508743 (218 digits)
600: 39937810748209753430...83373172999304809224 (263 digits)
"1 minute and 23s"

Pike

Translation of: Python
int MAX_N = 300;
int BRANCH = 4;
 
array ra = allocate(MAX_N);
array unrooted = allocate(MAX_N);
 
void tree(int br, int n, int l, int sum, int cnt)
{
    int c;
    for (int b = br + 1; b < BRANCH + 1; b++)
    {
        sum += n;
        if (sum >= MAX_N)
            return;
 
        // prevent unneeded long math
        if (l * 2 >= sum && b >= BRANCH)
            return;
 
        if (b == br + 1)
        {
            c = ra[n] * cnt;
        }
        else
        {
            c = c * (ra[n] + (b - br - 1)) / (b - br);
        }
 
        if (l * 2 < sum)
            unrooted[sum] += c;
 
        if (b < BRANCH)
        {
            ra[sum] += c;
            for (int m=1; m < n; m++)
            {
                tree(b, m, l, sum, c);
            }
        }
    }
}
 
void bicenter(int s)
{
    if (!(s & 1))
    {
        int aux = ra[s / 2];
        unrooted[s] += aux * (aux + 1) / 2;
    }
}
 
 
void main()
{
    ra[0] = ra[1] = unrooted[0] = unrooted[1] = 1;
 
    for (int n = 1; n < MAX_N; n++)
    {
        tree(0, n, n, 1, 1);
        bicenter(n);
        write("%d: %d\n", n, unrooted[n]);
    }
}

Python

This version only counts different paraffins. The multi-precision integers of Python avoid overflows.

Translation of: C
try:
    import psyco
    psyco.full()
except ImportError:
    pass

MAX_N = 300
BRANCH = 4

ra = [0] * MAX_N
unrooted = [0] * MAX_N

def tree(br, n, l, sum = 1, cnt = 1):
    global ra, unrooted, MAX_N, BRANCH
    for b in xrange(br + 1, BRANCH + 1):
        sum += n
        if sum >= MAX_N:
            return

        # prevent unneeded long math
        if l * 2 >= sum and b >= BRANCH:
            return

        if b == br + 1:
            c = ra[n] * cnt
        else:
            c = c * (ra[n] + (b - br - 1)) / (b - br)

        if l * 2 < sum:
            unrooted[sum] += c

        if b < BRANCH:
            ra[sum] += c;
            for m in range(1, n):
                tree(b, m, l, sum, c)

def bicenter(s):
    global ra, unrooted
    if not (s & 1):
        aux = ra[s / 2]
        unrooted[s] += aux * (aux + 1) / 2


def main():
    global ra, unrooted, MAX_N
    ra[0] = ra[1] = unrooted[0] = unrooted[1] = 1

    for n in xrange(1, MAX_N):
        tree(0, n, n)
        bicenter(n)
        print "%d: %d" % (n, unrooted[n])

main()

Output (newlines added):

1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
22: 2278658
23: 5731580
24: 14490245
25: 36797588
26: 93839412
27: 240215803
28: 617105614
29: 1590507121
30: 4111846763
31: 10660307791
32: 27711253769
33: 72214088660
34: 188626236139
35: 493782952902
36: 1295297588128
...
498: 8905549466780876642396073343654258905917693028466999354614122414
334803973053490695667380301725459866905052509157458138657523790806693
604983304945043446074074823147488033014740909049433466213254508146498
309733349188106
499: 2494778503569518415974916458171385008434293725770646826239156067
242244971575380601975205121412154447546983792046199118690890088809311
353933980135961153586789990128720078992516401152141409189797471635981
1774035382416687

Using generating function

This is almost the same as the one in Formal power series. Compare to the Mathematica and Haskell solutions.

from itertools import count, chain, tee, islice, cycle
from fractions import Fraction
from sys import setrecursionlimit
setrecursionlimit(5000)

def frac(a,b): return a//b if a%b == 0 else Fraction(a,b)

# infinite polynomial class
class Poly:
    def __init__(self, gen = None):
        self.gen, self.source = (None, gen) if type(gen) is Poly \
            else (gen, None)

    def __iter__(self):
        # We're essentially tee'ing it everytime the iterator
        # is, well, iterated.  This may be excessive.
        return Poly(self)

    def getsource(self):
        if self.gen == None:
            s = self.source
            s.getsource()
            s.gen, self.gen = tee(s.gen, 2)

    def next(self):
        self.getsource()
        return next(self.gen)

    __next__ = next

    # Overload "<<" as stream input operator. Hey, C++ does it.
    def __lshift__(self, a): self.gen = a

    # The other operators are pretty much what one would expect
    def __neg__(self): return Poly(-x for x in self)

    def __sub__(a, b): return a + (-b)

    def __rsub__(a, n):
        a = Poly(a)
        def gen():
            yield(n - next(a))
            for x in a: yield(-x)
        return Poly(gen())

    def __add__(a, b):
        if type(b) is Poly:
            return Poly(x + y for (x,y) in zip(a,b))

        a = Poly(a)
        def gen():
            yield(next(a) + b)
            for x in a: yield(x)

        return Poly(gen())

    def __radd__(a,b):
        return a + b

    def __mul__(a,b):
        if not type(b) is Poly:
            return Poly(x*b for x in a)

        def gen():
            s = Poly(cycle([0]))
            for y in b:
                s += y*a
                yield(next(s))

        return Poly(gen())

    def __rmul__(a,b): return a*b

    def __truediv__(a,b):
        if not type(b) is Poly:
            return Poly(frac(x, b) for x in a)

        a, b = Poly(a), Poly(b)
        def gen():
            r, bb = a,next(b)
            while True:
                aa = next(r)
                q = frac(aa, bb)
                yield(q)
                r -= q*b

        return Poly(gen())

    def repl(self, n):
        def gen():
            for x in self:
                yield(x)
                for i in range(n-1): yield(0)
        return Poly(gen())

    def __pow__(self, n):
        return Poly(self) if n == 1 else self * self**(n-1)

def S2(a,b): return (a*a + b)/2
def S4(a,b,c,d): return a**4/24 + a**2*b/4 + a*c/3 + b**2/8 + d/4

x1 = Poly()
x2 = x1.repl(2)
x3 = x1.repl(3)
x4 = x1.repl(4)
x1 << chain([1], (x1**3 + 3*x1*x2 + 2*x3)/6)

a598 = x1
a678 = Poly(chain([0], S4(x1, x2, x3, x4)))
a599 = S2(x1 - 1, x2 - 1)
a602 = a678 - a599 + x2

for n,x in zip(count(0), islice(a602, 500)): print(n,x)

Using generating function without OO

This uses a different generating function, but also demonstrates a lower level approach and the use of functools.lru_cache to memoise a recursive function which would otherwise make an exponential number of recursive calls.

#!/usr/bin/python3

from functools import lru_cache

def Z_S(n, f, k):
    """
    The cycle index of the symmetric group has recurrence
        Z(S_n, f(x)) = 1/n \sum_{i=1}^n f(x^i) Z(S_{n-i}, f(x)).
    This function finds the coefficient of x^k in Z(S_n, f(x))
    """
    # Special case to avoid division by zero
    if n == 0:
        return 1 if k == 0 else 0
    # Special case as a speed optimisation
    if n == 1:
        return f(k)
    return sum(
        sum(f(ij // i) * Z_S(n-i, f, k - ij) for ij in range(0, k+1, i))
        for i in range(1, n+1)
    ) // n

@lru_cache(maxsize=None)
def A000598(k): return 1 if k == 0 else Z_S(3, A000598, k-1)

@lru_cache(maxsize=None)
def A000642(k): return Z_S(2, A000598, k)

def A000631(k): return Z_S(2, A000642, k)

def A000602(k): return A000642(k) + (A000642((k-1) // 2) if k % 2 == 1 else 0) - A000631(k-1)

for k in range(500): print(k, A000602(k))

Racket

This Scheme solution runs in Racket too:

Or, a direct translation of the C entry:

#lang racket

(define MAX_N 33)
(define BRANCH 4)

(define rooted   (make-vector MAX_N 0))
(define unrooted (make-vector MAX_N 0))
(for ([i 2]) (vector-set! rooted i 1) (vector-set! unrooted i 1))

(define (vector-inc! v i d) (vector-set! v i (+ d (vector-ref v i))))

(define (choose m k)
  (if (= k 1) m
      (for/fold ([r m]) ([i (in-range 1 k)]) (/ (* r (+ m i)) (add1 i)))))

(define (tree br n cnt sum l)
  (let/ec return
    (for ([b (in-range (add1 br) (add1 BRANCH))])
      (define s (+ sum (* (- b br) n)))
      (when (>= s MAX_N) (return))
      (define c (* (choose (vector-ref rooted n) (- b br)) cnt))
      (when (< (* l 2) s) (vector-inc! unrooted s c))
      (when (= b BRANCH) (return))
      (vector-inc! rooted s c)
      (for ([m (in-range (sub1 n) 0 -1)]) (tree b m c s l)))))

(define (bicenter s)
  (when (even? s)
    (vector-inc! unrooted s (* (vector-ref rooted (/ s 2))
                               (add1 (vector-ref rooted (/ s 2)))
                               1/2))))

(for ([n (in-range 1 MAX_N)])
  (tree 0 n 1 1 n)
  (bicenter n)
  (printf "~a: ~a\n" n (vector-ref unrooted n)))

Raku

(formerly Perl 6)

Works with: rakudo version 2016.04

Counting only, same algorithm as the C solution with some refactorings.

Note how lexical scoping — rather than global variables or repeated arguments — is used to pass down information to subroutines.

sub count-unrooted-trees(Int $max-branches, Int $max-weight) {
    my @rooted   = flat 1,1,0 xx $max-weight - 1;
    my @unrooted = flat 1,1,0 xx $max-weight - 1;

    sub count-trees-with-centroid(Int $radius) {
        sub add-branches(
            Int $branches,        # number of branches to add
            Int $w,               # weight of heaviest branch to add
            Int $weight  is copy, # accumulated weight of tree
            Int $choices is copy, # number of choices so far
        ) {
            $choices *= @rooted[$w];
            for 1 .. $branches -> $b {
                ($weight += $w) <= $max-weight or last;
                @unrooted[$weight] += $choices if $weight > 2*$radius;
                if $b < $branches {
                    @rooted[$weight] += $choices;
                    add-branches($branches - $b, $_, $weight, $choices) for 1 ..^ $w;
                    $choices = $choices * (@rooted[$w] + $b) div ($b + 1);
                }
            }
        }
        add-branches($max-branches, $radius, 1, 1);
    }

    sub count-trees-with-bicentroid(Int $weight) {
        if $weight %% 2 {
            my \halfs = @rooted[$weight div 2];
            @unrooted[$weight] += (halfs * (halfs + 1)) div 2;
        }
    }

    gather {
        take 1;
        for 1 .. $max-weight {
            count-trees-with-centroid($_);
            count-trees-with-bicentroid($_);
            take @unrooted[$_];
        }
    }
}

my constant N = 100;
my @paraffins = count-unrooted-trees(4, N);
say .fmt('%3d'), ': ', @paraffins[$_] for flat 1 .. 30, N;
Output:
  1: 1
  2: 1
  3: 1
  4: 2
  5: 3
  6: 5
  7: 9
  8: 18
  9: 35
 10: 75
 11: 159
 12: 355
 13: 802
 14: 1858
 15: 4347
 16: 10359
 17: 24894
 18: 60523
 19: 148284
 20: 366319
 21: 910726
 22: 2278658
 23: 5731580
 24: 14490245
 25: 36797588
 26: 93839412
 27: 240215803
 28: 617105614
 29: 1590507121
 30: 4111846763
100: 5921072038125809849884993369103538010139

REXX

(Based, in large part, on the Pascal version.)

Programming note:   the biggest concern was calculating the number of decimal digits   (so as to avoid integer overflow).

/*REXX pgm enumerates (without repetition) the number of paraffins with N carbon atoms. */
parse arg nodes .                                /*obtain optional argument from the CL.*/
if nodes=='' | nodes==","  then nodes= 100       /*Not specified?  Then use the default.*/
  rooted. =  0;     rooted.0= 1;     rooted.1= 1 /*define the  base     rooted  numbers.*/
unrooted. =  0;   unrooted.0= 1;   unrooted.1= 1 /*  "     "     "    unrooted     "    */
numeric digits max(9, nodes % 2)                 /*this program may use gihugeic numbers*/
w= length(nodes)                                 /*W:  used for aligning formatted nodes*/
say  right(0, w)  unrooted.0                     /*show enumerations of  0  carbon atoms*/
                                                 /* [↓]  process all nodes (up to NODES)*/
          do C=1  for nodes;           h= C % 2  /*C:   is the number of carbon atoms.  */
          call  tree  0, C, C, 1, 1              /* [↓]  if # of carbon atoms is even···*/
          if \(C//2)  then unrooted.C= unrooted.C    +    rooted.h * (rooted.h + 1)  %  2
          say right(C, w)  unrooted.C            /*display an aligned formatted number. */
          end   /*C*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
tree: procedure expose rooted. unrooted. nodes #.          /*this function is recursive.*/
      parse arg br,n,L,sum,cnt;             nm= n - 1;            LL= L + L
      brp= br + 1
                        do b=brp  to 4;     sum= sum + n
                        if sum>nodes  then leave
                        if b==4       then if LL>=sum  then leave
                        if b==brp     then #.br= rooted.n * cnt
                                      else #.br= #.br  *  (rooted.n + b - brp) % (b - br)
                        if LL<sum     then unrooted.sum= unrooted.sum  +  #.br
                        if b==4       then leave
                        rooted.sum= rooted.sum + #.br
                           do m=nm  by -1  for nm;            call tree b, m, L, sum, #.br
                           end   /*m*/
                        end      /*b*/                     /* ↑↑↑↑↑↑↑↑↑   recursive.    */
      return
output   when using the input of:     600

(Shown at three-quarter size.)

  0 1
  1 1
  2 1
  3 1
  4 2
  5 3
  6 5
  7 9
  8 18
  9 35
 10 75
 11 159
 12 355
 13 802
 14 1858
 15 4347
 16 10359
 17 24894
 18 60523
 19 148284
 20 366319
 21 910726
 22 2278658
 23 5731580
 24 14490245
 25 36797588
 26 93839412
 27 240215803
 28 617105614
 29 1590507121
 30 4111846763
 31 10660307791
 32 27711253769
 33 72214088660
 34 188626236139
 35 493782952902
 36 1295297588128
 37 3404490780161
 38 8964747474595
 39 23647478933969
 40 62481801147341
 41 165351455535782
 42 438242894769226
 43 1163169707886427
 44 3091461011836856
 45 8227162372221203
 46 21921834086683418
 47 58481806621987010
 48 156192366474590639
 49 417612400765382272
 50 1117743651746953270
 51 2994664179967370611
 52 8031081780535296591
 53 21557771913572630901
 54 57919180873148437753
 55 155745431857549699124
 56 419149571193411829372
 57 1128939578361332867936
 58 3043043571906827182530
 59 8208615366863753915949
 60 22158734535770411074184
 61 59858097847706865855186
 62 161805725349297357221898
 63 437671691526158936922623
 64 1184616185385310843585573
 65 3208285066181475821271463
 66 8694130712024868414002815
 67 23573796134448175745408811
 68 63955159527348138708694312
 69 173603007393950249896865875
 70 471484798515330363034639871
 71 1281151315764638215613845510
 72 3482965749140691245110434511
 73 9473447386804490449091871124
 74 25779306238954404972323916397
 75 70183211512214096492433058105
 76 191156381393249393027319384769
 77 520874195248906781713044332539
 78 1419908915343952137338409797325
 79 3872282575137005474139119076135
 80 10564476906946675106953415600016
 81 28833609436277333169440806135431
 82 78725585464391037293036629979444
 83 215027809474796675607407513633870
 84 587531723826577193455385789266377
 85 1605913778494711520354663202536756
 86 4391002908093323425994602631972445
 87 12010257907756938974208750945664835
 88 32861295558120887536942123568548502
 89 89940959024891576997396491928932689
 90 246245150242821439632304475956113295
 91 674391606297983432514229725117306224
 92 1847515048012613337782670842346319120
 93 5062818112121161180862827915688625902
 94 13877857529584521384324419956411729295
 95 38051836070803837001309074456088423358
 96 104363664561059273927704242814298678658
 97 286312976836850192359345859166390622180
 98 785684759853087702778573182234297830503
 99 2156596319845084996862701478402986311496
100 5921072038125809849884993369103538010139
101 16260750014333666174953055376699249561110
102 44667063168726812052821334495766769690630
103 122726610195426301690448676841677827340780
104 337281538963751874669853952178948219200633
105 927143441542280244466720172757699607129825
106 2549176520305910764377448963173035784835631
107 7010510656300876673813654064741809461787182
108 19283877336110239907079044091958051661009951
109 53055727810105880736027950213934519705620559
110 146002972524313232817393491844985704938385801
111 401865724190508834753025926637435418813476039
112 1106339625432222709435767174129826811545391101
113 3046369875968510015403046201590835240153395100
114 8389999420170754836800638580300552381250693062
115 23111326593011774543116815302964652139347135182
116 63675155467360117136901070528242608498818046250
117 175467195960062612437322237574246321515725845634
118 483616671898832299071277369263305813784565460114
119 1333167312321418940566764416056977442040495550342
120 3675740183950426011078357941139728051663026172228
121 10136322901774027447848977748665383292736169662267
122 27956983197937526275999613945221497078279509595407
123 77121096978813982358935411851692069578533009193138
124 212778592638033483022781655638827961970402357080215
125 587155794584829621068447884048323985962957796104395
126 1620497362318232091081355117667505915417499978679013
127 4473132502312622821884079561929897829404710575328024
128 12349306792492607803837161096610238756912653878568775
129 34098849774876383478036291434385545792965491914980650
130 94167748474814466028838037996326649316233175269577493
131 260093170948828891650104553710684162327855828421145690
132 718487205759724277833835055443909476145495116155508155
133 1985050220521088907210323840127550973214943015739291120
134 5485110653386099899275645856977233423965042141295771502
135 15158624968755754576600389921905653584106659889930620820
136 41898053824932615440265041900412507427220728337249680527
137 115820822448502452349822520317304132018285539473087897141
138 320211802888589798701825810680319271475504997973219083170
139 885411355238188116465394365370757710295372148438998022826
140 2448550585524918609600214967948504177437555812600018440773
141 6772180336728084537425567078328320492989943976904644119200
142 18732796033402227075307540055538834651333956120072379687678
143 51823958523558404531622255138725304201359354985976024954747
144 143387634030485523461662580179416231260007790242619239696168
145 396775836020295064920040342935953476579230225268967120945252
146 1098070975453594757891511609218085888434254839495604326720679
147 3039251105982158526063018965393900608357891201531016545453671
148 8413041613874240075848233530979949485087059914285837491890647
149 23291051500594069758631194545655502320903778728677961917787017
150 64487285324785805685734825467573942213924157583096655274158296
151 178569541961158786360447600422369518262867694211679827307797522
152 494525085028771691070376002671999818542495469214503552543494392
153 1369671107847363840368349527801907625890550280754690871159384167
154 3793941909035282970536126899217159922244816989321008990552343933
155 10510197366726219419291185594221700080820692107164072063617583537
156 29118988780427095392911694296588496006042150251385271606702314123
157 80683801316548713731547508195369620842564695928190934573122040053
158 223583881196691039626561929582827819978293196688915654006262185620
159 619638153674192054430980737083649826660231642062541457264064352590
160 1717428978037773119953669826811378686605009454833429087085211111817
161 4760601845949152288761851253868434642647478893273231351009713026471
162 13197353449186709568929592710369551730672357721629256541119049584662
163 36589226826424289787166608629764201634396432211204018812233928108927
164 101451975263926040804307438557581821336425438886780992752450611791029
165 281324901033788598583154170205263556814795889090791609969956549076553
166 780181677818281299965193432627955631078491817302716837810578348379410
167 2163828038323756757063639945018570904120396578192324738853110253083851
168 6001899167570139611127915072874671685163847392112466633395193150607161
169 16649191065671323727576273232609293462550308875754570697371028619095529
170 46188686972056579073145176280276791118176099297131728121378351698216964
171 128149137125681447665302588425507023489080631426025859378879991574150361
172 355576383032176188837060897590191000068058505378256432541548821711409736
173 986703913063443346422020725722084251185909113284392827422830038792419867
174 2738275183964917202164682060710234556685852044781624370789938274187242387
175 7599818348156354735525837090092498330135165342551619766604085368593605623
176 21094284799140605474267783653778252494175708547669907184929527663028371844
177 58554660677719531288883019197284429180673377561888244491058170393359945984
178 162552183133868639189244204285356619593212307470997836346642760606493409411
179 451292826786530619879633220482642976940485477290114448603416892241141577694
180 1253019870825476025726441067676567248038950763298814178748038046446512128926
181 3479293084378459187212303139960535018989517537846033787292960498791544468857
182 9661781855977284524013799278118239872342899149756408496918889491272019198160
183 26832197158239597797570968340612728947891256166650480273266227097169558934791
184 74522545727244539603451333395337695567614066525612042720018110555143893455632
185 206990881176753531116559188573370889805581604324744059300494333307748123498957
186 574971719221297425559348824161112797452658996937464320320048053830834065076638
187 1597250942564001477500533605167309927398304330031144648098072721512668593957703
188 4437423571982333312534972159110678450135834859468229274326790786916695731276497
189 12328758711422329105019389982539560951228986597668702145097956175468519348920309
190 34256124585721478074980980873512523523896822875906637442595527046990665266761523
191 95189094589104790904556217884090558824685828617516319665565748564984723369457220
192 264524521940855272106937702820301986845470010150944446148610489622177560655580196
193 735146927788110318878638054407335543366855876665936464594690408421993895145574507
194 2043202995476015462049187462882169976289343296164934404442378707876446055277665852
195 5679076882963913929265887525377096781591407289261632655627568444557125995319535956
196 15786016263625679649343179544010857226174369384245579060916786714528288038933116607
197 43882930188633901470015828734437451959746697520345645823789728504039719238235284266
198 121996306076853365751053531202168307620916572983606780123661900035869303555630148063
199 339176261988518728096836182493660862745709169352281541101577697702699073887422989905
200 943043328799038505167332910595466006794464252841664581909549826351576307818857723954
201 2622195090600379263364346956264279702121691087227848066895838284611152725976467138514
202 7291640328972323818932818268921088199080628040707037288217930491456875016135548131376
203 20277391980621940663950418790370236703345679504035237316723532280155012704421841134349
204 56393014827755686247101145333945229562531368138401437274854961321951321148392478837326
205 156842815530515935964014240194651333844507609987619166694876840912879248319077130581042
206 436244327522179535577207667646065280269833187002466982658692809627210486721781255271000
207 1213446271931548955557154166653292946893343739485414159573064525913091105471901382363618
208 3375488708820014134062868343953409434477577207616345786043331438445002602958236463369496
209 9390265533842684145381903993662706957889355090166833985181893569515242461156430420087174
210 26124257322713604151166532772505893583948958402141893219135619374261694917895951421995216
211 72683304203243676344755903584747211387194000236278332965871882088118015556154063840306823
212 202231949421481999766866699910650758032171534187352358158377153614156835833334273717131223
213 562715711666310319461011612553561742990998466225938804278692337156743603187662151365368333
214 1565857565336512188705390960430387447731996020954944548096469264981742312029607101653583428
215 4357517959671123300838959993742696621943017700847213254885626681813005595459240045099075216
216 12126894898610872886310565416845280412742023065548437498642351657251934710541687754081654026
217 33750741717021238734330907104325257824118779361033241504478398185637818043649623474315265399
218 93937737312335248803818931862078208076074752854934195324106075781489944419493627198038389700
219 261468709433838684317888993242737209511093586216774554328802510011968600660643442350819036063
220 727816668798656458204462998706538701005731200304158665308881919140152438032334192310185418095
221 2026033924729657796058178057776349080819410108838068133330231127278657184350600523917658261950
222 5640189120704586237460028096950040726608416653216674308167097595939322372553028535058020450921
223 15702277858615709125768061794209929881772942003228681661684701314937614138722747596565403037124
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467 121175387073295736524569718822097639420463861716971063492109528962492569785076732921975740253331081548338333727361459218154437136082325290987001985839144830920858486257553315470385995502530184051873497066
468 339344840189475536421125940700761759628752222992160549381027643941745741914656836040232730595608192347759637830477587902350052606592456881569785539786928879480686510907084881467566111373965693128772815168
469 950326942647405347844160146627862711768881566137164131522937150987646591788240557198082994780294550892371568822419949180800659521354470469743632592095655784774066429590870707610663947413928541377601304331
470 2661397660073852552129723028095885797019532580891659413666656327685520361494990644244780459416494337505306325842984790687846760034636266016838667025621454787635756967010007792781115241634931034023342955909
471 7453348271319000807792642584688862626586099412065267712592599510401797049893846344817387925831913586906215688069450852737050111320975144949815225427831362617279252934288022295734101068785815612716970311808
472 20873628693465442369994091580260968702372775440196031927070687104083645990021388958800099647202094094135845379561614782887105082175039614621895682762293655052254837096492726532427207985365110705700858185105
473 58458728731766254613260704634770549014535819580818340519042652762258547931670198980355298780573863556078114808742211556195563024034089763095349562343593909106562326371091563375170455486291957378525618197831
474 163721469153385637408150476563293561192630360744595190990592087993468898713335977248677815933276982928116057236035906891546690927202148725585574081941714037659737484743937984394080547704511778110792121687207
475 458528920077354685502169678831532604357019434399108847894808896709351443638466718242161166503349046639203448149470195902534259098807880444564985003619318026651644088082419427990684019605853227416774184858732
476 1284199937356117010353036111976291463604172133918493472823892597653505936748633559438891259654067719601811837856803850228207178704881159679844950373205860758636370584332260625151603176391842907361800288702852
477 3596692826870076496488870100810883119867190535450061805109321868771960959615817567820544952252639528684637338431882866065151676205653648613261366208363844491931760159185397360397157081159998934076146625574324
478 10073463693660743814830768135279278537079909784776055480865370746882390281411517464342747341359496246287190961417525136257220264919874937072794203921533179745349010643055354298890387265997799866799848104338235
479 28213635719632829111553766945298618684025617913525005700531345253000364443673781640156844004227127409788826310407619909467100939134351426418260762969996556114029193750675871742395710571704344996337919334627311
480 79021272029812529198306048278738451404456552316178623835088343424900929896529145986849511125817498577910188475147844537607285370369164399968397118901967107253074866629088624713978766067247085915460907203507916
481 221326640007559663668965755167218010334237161338270139368837726672890339102089097625270467903768037212482685022578414146878493502591843464527824997936161209273201301931996307060804297022524160462260698613537211
482 619909167426938833226794412160056195934133884103637737354763534557844211624630276625508115154691754605800841672755247108234248396654245317254494351293449688401442053466436766136865155607831934082917347856426393
483 1736309336940064055314375159645968597229131182622025720281597392750374289881332068277426029158100061555043458510124867482152338705144101468366800180174515094346512495259376551669075968372524272389756751432854853
484 4863297048468267534636166312939910470952766062828082277824505171156300282060030581670231761463878726184462085106767555905935986480934403918197493275089839501765798644464700331235805831873720464298566133914310111
485 13621945190301700706138963992367401939405566877527256386624696446539344800813487799938840805754463459615136581585117531096761299146137690070987761802590544736755857013785633546389925702052333218769869982790548052
486 38155054329515229613775470033224877184200699251781861664268079502676128352861614648198071275587050390975562955844380056803299861976593575165303651531182196565505294625797093250325990896145553936059099182468469941
487 106873398849478133039914184580874875231786272245238762673187219575665471269978868235613615591616484938408255629005435265191765938106429802557895876693594503003511545832633156383000727183561568902367863186683667386
488 299358603826075402277899963139952216438074968330347955827805421649062237131327956205363209204249356420478800321507971357294900909553107391222810893361186943113207800413096895718566298784120554090436875783734064658
489 838529657705833034226462835006399865426986819335499347214210855084885678859732280609017110745548560768471886696342486720814635172359590192655917206826346091436208169907230087809358382454265923070116812677298151128
490 2348819542369503943745770103735793608241357602923831591641116837956213501074463807697474203912578951142821017553783252607800118192578200768970029556421579893140989343065980349644833804994551495825731452145526250765
491 6579386479996359380652505811548331172430892668901276541544233471990370985480393191871721148991766536717220264862961217017435585211918629713359048858315079558167250616463096722818683439439764375707901662500150442223
492 18430013297106462705563823748850922689996624541574571380773518214465413984005545896077127791090690118569929503235833070046114981151702312880527866333379163449235952921156068688050620988293428148558907616760764230182
493 51626226783162901884788448815927116139784145435568714116360575592327600891647514042914267976366137939193067990947983679977241677448127920019074746593823496272420143919124413908756022387549336194531782497745210053318
494 144617080122147701980280207133464988259100003413798968327225643978306456666956314890639498978440890457630421021855428539194342433864427806316257734937387813993364347367110868522308183407058843998076264751693102167102
495 405110259662180001404402330891745929672485944541334128902338780325351363683988833428262927776387719886899607701108769883875334759722976317523430427909747225193367590120906316894975981073694814441448542441371959370139
496 1134831355904717012865001274054429116248775025280754969468716824949667862347555347996290225366000942364709536217262581087333644261895222727600153755613494310524232705702001879535164138079533205048872360533370572443014
497 3179024139613983878848518783709510209263571759925674510120249336292840845927230735400333746765269611853557998210338782704187073085423815480824362906609395745543530469335794431219957388926521344251394881172155468218720
498 8905549466780876642396073343654258905917693028466999354614122414334803973053490695667380301725459866905052509157458138657523790806693604983304945043446074074823147488033014740909049433466213254508146498309733349188106
499 24947785035695184159749164581713850084342937257706468262391560672422449715753806019752051214121544475469837920461991186908900888093113539339801359611535867899901287200789925164011521414091897974716359811774035382416687
500 69888806977968318652007568872323509883145559195919337637376568556764896550639165374194198834566064897188044072953504164443147167221896769574954040005634802439024034052447587053721967608928388404735140565376686372508743
501 195788691591167358983391697449037971940030264992640916356261226388550228184090102380273799313853791372688026208122257167117578777323459026117493349556753561788856363822885976681867205634092580493325710364904047588782531
502 548494032946436495469620611795432979620763461369106146549869347798577635070861996537588958985456038738490074623264997230333113526957977331005103506330698269434761435756409311208727307430024569502936206334553694283676634
503 1536598904504020022390182411725047622200819847186681809932497847822676986046575419040245980552330718581280600217997983244508065171419798533850912230390660880590427418403620514675655804742184351964865010834911870260532737
504 4304804399696237237507463761025719027368507839779837180796834025304374000283519921714975073048412322678796133355944724703959888901927384437117293324439590274617544550675650074944617797896522994225668856852390729039132854
505 12060091457367838356213610817207234492788918818485067716307605848734397401969983569015491356977123520274608350782427421607680673607611421693067241201544491744636061545720805042074071705277143768237509563975581521481971744
506 33787187151401359474138248785114344113312671816443199831447300383553450708584370216211336875384723399889706123181744528180234956967328946497287340103298696637196990369343041151384152213862740349556350988246854939570357460
507 94658085087600767191814295625767249595924998256072418807984205214517757901371372122612009197699848176311901481271194791454722663092510105871357747683604714671675892751969983638228197725633072216463700392042188326488394134
508 265196394876068538373882629260283401265291216082994401252529458513871580318500770697711905539654827042665184414463166697227923698966383149290066454899757132953798122218684163429442520747546194624074319341533809187955859850
509 742987872327542939782616463958400742805603198481051214041825574666488856920606250928581874241724462792461629424623763690623204714770029065303797191854683567405043220191095551009306245915236192966672841365430534911831131941
510 2081613160807052627852671106429699815346637106588558039549850943282344854796785381431301194195553809491424445009953316091702948592825818399878585803321563802098469325842437856189296223605845914855517526617036889123791916879
511 5832066935792974884467907134920571271897853778052470885492407657804476350992729502794437759057956535540384995556877321698401457926234137392312868296040813445628205937149453084385402716294640357428781165896966449684574454361
512 16339890167962447114698099986458863352206177083383900932997918005501783478563092435529831406801065305291687696314783303059532484613725570916382493884546624303413507269714944392090534814565582831222233859429586344081533623241
513 45780434299714465900528472125920725498064971363403795279045937481470113424325181182505262687930126114077659876437271891333711756627584325630066079172781399287862623328125519649485377842290310077120775093950497665089366130687
514 128266962310953974817496160900699691497973827713233238862518018518153845134405137336599725997863436033386936752436933102935100419014549625516239149651689776363494280982666232916940899586072745348967003932355195829652710565323
515 359379930993429264828841632427244309223147010714116927910718740188846797318879347570762566427252001469939998654930594986004094801299842961112093140981896788981026626501686993621027437750157900308470012337112622156560636947110
516 1006924542315488374500853560606537597587628897577216253321035732074520962650385695695957162041323200214214989456931729021525131967461613489483575393225063950698261368046789785512490836183019178363176483284998222852131260793329
517 2821266485261678190970175585995599013302402666565073477188760415313469345059253072487099276926742680217597162061717288831991636305900535562690589744681682822209861817463637168946441775672540557108754665007706181417194852743673
518 7904881349656251061473987277087113441221261735694942390935942379504114520513016461328765196760949216247834315180133902874405614838577638980548097603486286719488897631013013645540201605430865161968047603658933276895894118126577
519 22148822783476459665161239520326747892863972254121722623165980783450639660107999871629236134994978017405408286812397246138293987458554447735277276332009520637564097247635625196663974747191523216939685776154127668063344206727515
520 62059742901425904629338150724437488232533191831824171835399430762211865699881006074035063307010595844284819123816004126700235429376699704613442238250638578034471580678600448677946228171430160413252521269033173497010829305310070
521 173889481103654993885697912180230847109956571064408633157840864019960668444788394597848388174035112832058754118840959218604629363187380115374538541239969681456462399851049935709259430469140038092416800099936809837501548629419251
522 487237438353510461480322055175216685358453440965503679089110259244098601622481946525987154583247290474495825110790428232083837065845141230995208072661400874125847173599395779442881348010764780130289373675407844748825375982291576
523 1365249341977970463286629534436802597874909462165553276526036533935380410982365034267951544109405793394976898112548175325365716440901089833896913384181543565844061890207485518367244573267650254673044598876870761459673032426572791
524 3825491754932762419915561097537121564200743271411781620746512306663469742497358308861993572741066210926122399635337243791511129258811126187919571241817605904443010401346719713702760894877466325186307517614731894445947894796767114
525 10719302309878652380384516356382166639006440998188516638004132546072910806665312028794505327365735543508251609975945836859111394467645220818489365630282327075608186191418644287250242839490280434025629012415174874427233644792719126
526 30036526459402062339463395705693696485866208596205236429655914914106224348466112590375499803794725589039703740454167809009734970232956535563557834120671943961554465487325495175688210953272246053057252063214625992519724954037612961
527 84166025782723224095384188649790709148610075463432507426660325284564796705881224942036706561317071843116699201695872922866817921335289864428901472097479476719671087269605856347371997124934383552502396699362196988401346430646747006
528 235845635416842984456760777132945515856999196345706521389832909973935764210127671873134402454473246185921537102807505090269863242522141089414715962011039922723491733664638304673495569805055060565311063722326422751154200368631923503
529 660880231379418802777405081733926492570382106194676639514531412541032743121901858634475519373238487979339786507626766650356659413373515413279306429681912200932474828848682189973033135004565071892737745753906027827392728494556608253
530 1851917173296694678131177893097318031623351289196824232764870955345929906799243777446150541763007544665282692010482438584242343759620431067600724503315950171790180289180547484654739221490877726046711452233984632302850104520393724054
531 5189484539024974023060875229774459591175978879639211085596012701140080697953109311482167882706819672027235879767680334343132237433286607705575186891265480211543746643225881080885049561084461898859029632641541226282584718381268175284
532 14542220879050360468056495508039988477537129558157932975694199470190023232596163271638005095632655623681745060401562812445232701439964105915379192018753747780606081564529607769212580629016695103746761378973852075134397600207239056838
533 40751264346923384786996965333315471839035647346221284958868668144487725863041235362546027668216008491814785443004899010486870681224741576355160079456169818039399521103425839936168704536127581475054701011765204921022680006656506731092
534 114197148770541289129501657935967570120831361511197908446708083121615458350973234981741703062009747904717372645816351932083843518152909894099264366138522196870140181674523295392341523113827395326268784477436847600723381666850120301704
535 320017141527120548155297412500771871637046169922477155187755780091054695713444846561482870121158885207500121060459631775307006125397162661315301295022918385533306500201241016332699288226137659172385531665873544082877183260065443528563
536 896798795355927088583924740422419682680866663505270431857216104966363348283353096966112641466720640187665375866913775151778000064499549604147786887710103094823085924825063852439000077995374899965278521225209620423338100208823243577255
537 2513162496805305924994652929028227337900741303359422401258394445587620033563645371499586182646472574653524190122869992411537135046049723054704744895499809154948562458828371717215723933400837612554282072082923689324233263977702545335851
538 7042873528789804549023009429718508551240587395855454959447393625342687774113800208261550606915863764942627597293039530062620372289271654245519428805971979154263451098064110255730764962829489726948583677142406488354501935599650602704947
539 19737082687673556390721352264345841632628535899939477532501654872496250958883076796995396819937848508128319714087260948961412388238084295158556424245451789499629020892332182754145418430933844036596506680814926680881314886271496487397407
540 55312051809619551602032016880706543547044952891600243538996187044388577634128367035421891948845733143229655367153402733360285007971873542860702416019445927084453904646813876592387963001229881246292976508105361557214351135048699246219811
541 155010208742753427903084060324542228423870932489178221998163352879931170991350276156041821048277685181745196403821699135825031110010570548889285568859124774726102729888006672041845309890610453251798090166517895683828928695809150847161413
542 434414730403660666242480686151710405338291484348230483585265286431580631267787441681318931026972101375430655228073093667316312547851635688536686416789104596367223363266070786304645414433814791288332340665282553670531128654338935303062710
543 1217453777306053121974148130743827857250659905273978704087466955307958571108154566599556032709858439461424561321924541748674579082931332082048534393583969471373077566005250564194285092637693793477193025827897246702506377676667110888003051
544 3411961346304902166996546829236508375915661135096426152507382266780566115714464209731679505255350530297089589812760894910710662186652736515101681004796684951420075002848722945527385446694615063405612261948513249172344498476049032727368932
545 9562234557956215685447778971326839077971742508799397524726224980913770107542561233642537144755707764084441929855861338353411285576179719342733844867177863883572857211253257918657829949918268506597671210470859354038320983239803717948367057
546 26798984551944683896396972318203163300857663084556679915541460368274709450724851926215891854335257456859458212153626677811044926821774459229820661734053511910571785044003105044189638479606205266225446270840351353803273506169309471118208792
547 75107088400900935171428085766875460116078467071158059355550247373382656732937588776182338864603196235423355445938774847823000176252156320630305251715304955734307474521217572362374380356950458814490495201942851073681363411572726197723049322
548 210497597569301464288882416746271630627769460318548424487398185389081895427856735644325921548229743917952363214141905170334974750650776035004471387484354387432276219557447934141967586167564145736084901456735019380066463037900443368463750529
549 589952405670431312846732401541380055186678119288039300689019939966342269594068798076436018655794387803242115875612696777183707618498034665222357698402861839948544106654631760783965089206855637695677614562512652270011965573212912699150222306
550 1653447507782337854669859432138926115896644560678375475585702833879062465044870860669782613832197058452033490028732429312890629478909704586710851380721604876336726548704184794379564420818809125170496149758753891339593223873980466873657808675
551 4634121720374734876640015591275431220573719131896909689366804603883699372776723472237794656441957894804197374766817918202390675874135373912188537366950018228164738740572376864189332528537357341103370670904643559897426782873780924694404549952
552 12988172209958629354976479409887444232123870744472352461328494204014704070828026244185289280873080980631066297726926833848567305280970726703561881224727325697331320379610683962327265822755320616434222212741773936363823808982553652267656168868
553 36402583213297983893923717735314224806825009894247086975969745049876671625637320995335223140254138376863007565713896028681823975541880556241201015294981361924817665491211717706449102517872363313475464948800527281736167326461314836475762602132
554 102028128150871389624662995225471385626166470585516654271684749249146725344283355732835385231420923622121533361738287147345626269665983990682230430616181913975846190453144484442659897716807859861528370360657442232122713556070577420285383208688
555 285963874416020847109389094930639635310978070371917574848163514539256031622688475895043236807308840234385196458361427522717331753297967827774160725107466211955626920606425411335024033340578146198527828288903709377216287155450097973101513931048
556 801504474479813271309133372006480815774341489295873255358450479482345656539199652283517903763335961661569446982675025397282292830860672634283562452027344063269331309337781792056883435784253528457756731627198813595711933244967467670299374114046
557 2246488720207418329511276558277129470312106841874831124509410961099392817867249483985459588851395061897990263510836956434220351808930512362946163237936078660073482552772671903962215951191588803644208347288874453393266565443383162511657234118345
558 6296598959643092242021079585551078329850847268482171525470064761779801868460108402966668577295804134857293593759745117030878010004545736079682629323904363192618015053884473822270672571028721277391455244118991807746061002383902058418497875855719
559 17648642730266631868292372424977116267164237749404494221290915860131181882432797360164844699371892654424711605839763522329377096378132334711304905157957787701392734323958015971951220582183536590796265168463226513526448874376764310281387844265925
560 49467511648677843282150463393198182539660072909980483113248490098998257026549030443046473466516779500401900622056138457089583290673236119906693645839379735180063169334780902343519825146748491893711455668538850413590648271587545146478226087608088
561 138653960839720237780903529828886134179691393692197687961957505041394830362519491941711098742262499248612126723601317413164269527377101918721014795215731365573180931616856051047575192199455062578706819385718912289901845562979643787482220868470590
562 388640401404166726403604040849981573098393513226469303628837869996521785319521706543446407178328777717743373901923259217158612116651021164106312346635520946085334891999324129923894777900567248375001666785358736438720284544587738704414468904032120
563 1089349026371463401009368158207284707868138777281928593430252016526434574315276722928699429463879419586873781858260522963182861510174218509296474856977004106229622607819672469874387850085077891347597685755137100980144662073737174021186942689660110
564 3053441323563156856176173676476587746805206297444249491946613780577979509794372049192873666862100947279183913627112592173557057246256829321823438895271839470217698266648777542868826756796039640240841766865966822413852054415954144377331869096796079
565 8558852096820271775136720162461746547157423302243504173777592432887397654600479455464910907614603059047717971820020739888406164363436245821091289687550736029180777147833311398272790695986854037745909972157173521678102836203247861876126446623455941
566 23990807481507543363161119705635169223195483872753731231795993562626030819714639155074197774210088117109884822565992585843961866250403970561002968150135283942837164666422735496871886046653853601296395351635669654473996145870242254148487475307989363
567 67247725400514283441979337039015640682550676266572122233325347901774356639149525513033798919434539216083589575130443122562669309331966602929415277653012096623561084489778553727715607758071393441474715374180374812815472372801693951122572645561927972
568 188501022505615684981643115677699164754597350474195256056838146595177282144702175679897760845349918183759144278975953042018370336077526782303586476388543669978744010364213538970656378558771393693604806870918230917582142037940393284658036787796190095
569 528388293255699102768267980469032149927750544755411892116416888489381846695801228731476090820280927877733786235582677407755142595222187354458545616895456886946057362906637510923192846746478586898401249334809798543795790968101490955357548944689404419
570 1481139681819815984865596066119662646334970924597483028336367581391121951340303627468759478204423857870249313301338629009644743574956065854431325886381588648142897681667774313394610536579961171470177735442934858952609128677482450696112615071586842452
571 4151855116506668476194441700073079663723633462987242470405343562546713587958888648184810808699594843360967240724207789273501519226400493619246392208411552889455839912060087804590680797912338644051634046222148307123301178624036564900761321836191710011
572 11638357480345263990394090085259461185314769988960064237934836804925487110850655787138211592716050979856552696090052257943220978936791160054789403330241696199508450474010366551507612864466925270239152767446146366039050216218430677383776526207907402180
573 32624548811039863401037904007503284685293162205756830237882621136394392231068271280838341272733134688746658301784930533226029385976859586943837246137886034267907725333280479851554161785852662216907267193846654411745493408712732912484322747332569753453
574 91453565674803323768650830787276391811790063277550546004088694302628003182356333714047246018460967121395604723905367694772767795719531666793458510125166054851374267422730070214170914744385870914586163951159279595527449335944182886312691586553233544577
575 256365787336200929662034995821832390782901686928666874914069870370899411988421716154978333918736608690998878032992026547373667897661458006356779012690154706230243595830670415795736781249829251604726586760616697397361843738537308599671636128579766216782
576 718658845458758948702506099414021740112482114046088825658217733686793969462599685179137088238180952190394501163540929663698693688626584661285609749190141699964291053616343446908078388333027956836799572017270515282536504689083312961976194101462009359273
577 2014599660718172239358116605568856643151694289592073184425378273339989273348446339530727067696215854925213852206846102453349076412870365759644605659609783655714495452989015512852782223342828697700975206316701023605719615208342689777480895652276918504978
578 5647522879136413357043724962594194504509633277832144949066669490968478169978265556931566259002219427236007284245775753982950164107453385405960196151336248322201999888483166770542105522061775658707061620122677414898163155742057815886196812402259969187179
579 15831807173985862230107233053740741225612150185883101942540738540805830142634489058353779736162060565012145038054537756897243621466457330455955360553588041663278393105683723270458467556857755350606334738147209666024283469508953705750032998696854798882645
580 44381934001809692854756234269339315268470812596184703350405777120385183746883266152363516633876130125715212048430353040990333864666668831092958058599902474661246719890649315933844962255722663618054299097621520743489665201792451468614468913805975939827161
581 124418563451516717608159100241815057283891676377719678049022523196841786480992596032937228075126531995845396570532231756642367365743421776575398563615306941652795043493879881555454901683170515996805680559072186753736120076299373616446780086382772114931490
582 348792677349397309921528003866345863078288267128493115672623941522598776048251693732220755695431025178954502003908300766946990653422522409924817907661106508452087513434457878479779367729504982041734325073482182713922396135716196574656655917397176197508130
583 977806095585530428632890396349706483802108823441728751679077318093791239811258523588997322066823172053681534768862290564941339740251318834996032403685878595490338721229316118276427225932610752806672464800979531064027120225560291194836902693859291591304222
584 2741203745663437197736966440075137280851590485930314314587430580429575299512638713863969043150561831257032907715877401346996895296371290810703263836968320712939488378652866926641411909003873930822786738931046101321221325518315503962888376469401885265994060
585 7684808975913748652658967209307550026523625275725094128438468943922610561207598683967435518752918768168507923564618394072959860767215524485283100021926457609544261438530080673308562813716434922652310343666831692314313814970864451784428183836630928761650925
586 21544082791572906374946543781698049780372017958131916803345812602051605063383357232769995526919572102263994075038577092961024866879890540209628195098351235216405806648226367194569995759364031549796794238451414851144853052277666188839248681656548840911976649
587 60398493203123814085741196048964149458512214037197668589929833317931339018583950108268519326289514738313787604417699619567516222039243832183111178975585132940535614925187936726455836925903379425771368882224750816934541830962269930208271296475299926097625829
588 169327443006257573936646668349993867371898008622610164830955246279443424780534844823501342873267243826339934580783994073161973066663427908757748662682726366130882813806141273920032882868503108969241638154455553267567334572303893084865937522389951076285562747
589 474713668677088177094489482816227829504214352031323328270852779360795018627789537625498386079963131140368014696147596583604174811662296316215059880117728254587970371706740754712461820279546896203566157197971197495596350292834179333673528205132035438250223164
590 1330881086420173306472948842759034690388910835295708974701333347660281670372459868081546212221113168910799422984731503469641850319057297334418465703665451344306647829255276929453191448457762628094479367340404299426294764351292073296725138478070792305441994538
591 3731211685853046312563549804130192897894482665030016266213087718612969311730880556578228029440815256686371193612574126989800542685829595776767537625213336082451094407407753556113939616300924932057250588551086212328579069506182741208817570498197312000742027368
592 10460769517069739123687359810990316933489754837697607563060284873467767539631923775688564129175741466685148807938844948747499540214329998412416929773775341634383236367254529863598273698602704596025443271552973353883471455235251012798519258800836394435256519692
593 29327866858198207273574697780347354356736116274690885262860503347335873300288784621640105014427556311411239219193699064303605837210661734415346483263614357407858009798534956262057005804505509515500210664404703260756759631826962240762669956817971864561833592337
594 82224341953171929187993905974962398928689723174426292533299121654485229400096226495378345033289635192763405369603254224997947459074546576024079725573161809794749764393919071879732548335769111790225907188986528941931947672807840538706068520494757531062918651280
595 230527857674048343225923287378916402012939247063362423388045806625032470956505298314548628283027281141067660946488844016949754327695750825324231532143975797755830364765752272082152457678171775390388816889440647194887750291226936134708385231454852335253374241748
596 646322818871330819020026678859151232061945333744713991936981932978546339303692997493447216968561755944917675383810074482696023111390775261108373960890978770059536587716142486753445394021006709484586337320355689424155373151867240261358068356432336536173701295311
597 1812085231834964215986906286863558635801490208069682768154934692134797667605615863726859184203234040625899662283570285824836621732282277052641805198110266692539388281482866732375570715791640627557951307341871938752994799577907793806283669790186478849153786642552
598 5080550811840925643349676975073865244948497511823204612517164274084436176208486664180393520748149776872590054598605859910426770644290299687261305357381604813070954512351764834513332811381108395464045201934843255479950655358013934072061752933652412745207762928593
599 14244460781002294316293074748972676839717294390438741914842858681346369082771919307761594749385763933292062997927147254167820081532109568783310795522655235761093743035795930250205460385764201602882347095383486593446651566858974756939480439429601195323970185730231
600 39937810748209753430512819297411285489490181966942130366946252177280246828533288052682458081457090995010644569965651629864310293039821827595701938168213375114735446642304994845573628605370612685047772112154447318646467342856791534542089457868883373172999304809224

Ruby

Translation of: Python
MAX_N = 500
BRANCH = 4

def tree(br, n, l=n, sum=1, cnt=1)
  for b in br+1 .. BRANCH
    sum += n
    return if sum >= MAX_N
    # prevent unneeded long math
    return if l * 2 >= sum and b >= BRANCH
    if b == br + 1
      c = $ra[n] * cnt
    else
      c = c * ($ra[n] + (b - br - 1)) / (b - br)
    end
    $unrooted[sum] += c if l * 2 < sum
    next if b >= BRANCH
    $ra[sum] += c
    (1...n).each {|m| tree(b, m, l, sum, c)}
  end
end

def bicenter(s)
  return if s.odd?
  aux = $ra[s / 2]
  $unrooted[s] += aux * (aux + 1) / 2
end

$ra       = [0] * MAX_N
$unrooted = [0] * MAX_N

$ra[0] = $ra[1] = $unrooted[0] = $unrooted[1] = 1
for n in 1...MAX_N
  tree(0, n)
  bicenter(n)
  puts "%d: %d" % [n, $unrooted[n]]
end
Output:
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
22: 2278658
23: 5731580
24: 14490245
25: 36797588
26: 93839412
27: 240215803
28: 617105614
29: 1590507121
30: 4111846763
31: 10660307791
32: 27711253769
33: 72214088660
34: 188626236139
35: 493782952902
36: 1295297588128
...
498: 8905549466780876642396073343654258905917693028466999354614122414334803973053490695667380301725459866905052509157458138657523790806693604983304945043446074074823147488033014740909049433466213254508146498309733349188106
499: 24947785035695184159749164581713850084342937257706468262391560672422449715753806019752051214121544475469837920461991186908900888093113539339801359611535867899901287200789925164011521414091897974716359811774035382416687

Scala

object Paraffins extends App {
  val (nMax, nBranches) = (250, 4)
  val rooted, unrooted = Array.tabulate(nMax + 1)(i => if (i < 2) BigInt(1) else BigInt(0))
  val (unrooted, c) = (rooted.clone(), new Array[BigInt](nBranches))

  for (n <- 1 to nMax) {
    def tree(br: Int, n: Int, l: Int, inSum: Int, cnt: BigInt): Unit = {
      var sum = inSum
      for (b <- br + 1 to nBranches) {
        sum += n
        if (sum > nMax || (l * 2 >= sum && b >= nBranches)) return

        if (b == br + 1) c(br) = rooted(n) * cnt
        else {
          c(br) = c(br) * (rooted(n) + BigInt(b - br - 1))
          c(br) = c(br) / BigInt(b - br)
        }
        if (l * 2 < sum) unrooted(sum) = unrooted(sum) + c(br)
        if (b < nBranches) rooted(sum) = rooted(sum) + c(br)

        for (m <- n - 1 to 1 by -1) tree(b, m, l, sum, c(br))
      }
    }

    def bicenter(s: Int): Unit = if ((s & 1) == 0) {
      val halves = rooted(s / 2)
      unrooted(s) = unrooted(s) + ((halves + BigInt(1)) * halves >> 1)
    }

    tree(0, n, n, 1, BigInt(1))
    bicenter(n)
    println(f"$n%3d:  ${unrooted(n)}%s")
  }
}
Output:

See it in running in your browser by ScalaFiddle (JavaScript) or by Scastie (JVM).

Seed7

$ include "seed7_05.s7i";
  include "bigint.s7i";

const integer: max_n is 500;
const integer: branch is 4;

var array bigInteger: rooted is max_n times 0_;
var array bigInteger: unrooted is max_n times 0_;

const proc: tree (in integer: br, in integer: n, in integer: l, in var integer: sum, in bigInteger: cnt) is func
  local
    var integer: b is 0;
    var integer: m is 0;
    var bigInteger: c is 0_;
    var bigInteger: diff is 0_;
  begin
    for b range br + 1 to branch do
      sum +:= n;
      if sum > max_n or l * 2 >= sum and b >= branch then
        # Prevent unneeded long math.
        b := branch;
      else
        if b = (br + 1) then
          c := rooted[n] * cnt;
        else
          diff := bigInteger conv (b - br);
          c := c * (rooted[n] + pred(diff)) div diff;
        end if;
        if l * 2 < sum then
          unrooted[sum] +:= c;
        end if;
        if b < branch then
          rooted[sum] +:= c;
          for m range n-1 downto 1 do
            tree(b, m, l, sum, c);
          end for;
        end if;
      end if;
    end for;
  end func;

const proc: bicenter (in integer: s) is func
  begin
    if not odd(s) then
      unrooted[s] +:= (rooted[s div 2] * succ(rooted[s div 2])) >> 1;
    end if;
  end func;

const proc: main is func
  local
    var bigInteger: cnt is 1_;
    var integer: n is 0;
    var integer: sum is 1;
  begin
    rooted[1] := 1_;
    unrooted[1] := 1_;
    for n range 1 to max_n do
      tree(0, n, n, sum, cnt);
      bicenter(n);
      writeln(n <& ": " <& unrooted[n]);
    end for;
  end func;

Output (trimmed):

1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
11: 159
12: 355
13: 802
14: 1858
15: 4347
16: 10359
17: 24894
18: 60523
19: 148284
20: 366319
21: 910726
22: 2278658
23: 5731580
24: 14490245
25: 36797588
...
499: 24947785035695184159749164581713850084342937257706468262391560672422449715753806019752051214121544475469837920461991186908900888093113539339801359611535867899901287200789925164011521414091897974716359811774035382416687
500: 69888806977968318652007568872323509883145559195919337637376568556764896550639165374194198834566064897188044072953504164443147167221896769574954040005634802439024034052447587053721967608928388404735140565376686372508743

Tcl

Translation of: C

Handles arbitrarily large values.

package require Tcl 8.5

set maxN 200
set rooted [lrepeat $maxN 0]
lset rooted 0 1; lset rooted 1 1
set unrooted $rooted

proc choose {m k} {
    if {$k == 1} {
	return $m
    }
    for {set r $m; set i 1} {$i < $k} {incr i} {
	set r [expr {$r * ($m+$i) / ($i+1)}]
    }
    return $r
}

proc tree {br n cnt sum l} {
    global maxN rooted unrooted
    for {set b [expr {$br+1}]} {$b <= 4} {incr b} {
	set s [expr {$sum + ($b-$br) * $n}]
	if {$s >= $maxN} return
	set c [expr {[choose [lindex $rooted $n] [expr {$b-$br}]] * $cnt}]
	if {$l*2 < $s} {
	    lset unrooted $s [expr {[lindex $unrooted $s] + $c}]
	}
	if {$b == 4} return
	lset rooted $s [expr {[lindex $rooted $s] + $c}]
	for {set m $n} {[incr m -1]} {} {
	    tree $b $m $c $s $l
	}
    }
}

proc bicenter {s} {
    if {$s & 1} return
    global unrooted rooted
    set r [lindex $rooted [expr {$s/2}]]
    lset unrooted $s [expr {[lindex $unrooted $s] + $r*($r+1)/2}]
}

for {set n 1} {$n < $maxN} {incr n} {
    tree 0 $n 1 1 $n
    bicenter $n
    puts "${n}: [lindex $unrooted $n]"
}

Wren

Translation of: Go
Library: Wren-big
Library: Wren-fmt
import "./big" for BigInt
import "./fmt" for Fmt

var branches = 4
var nMax = 250
var rooted = List.filled(nMax + 1, BigInt.zero)
var unrooted = List.filled(nMax + 1, BigInt.zero)
var c = List.filled(branches, BigInt.zero)

var tree
tree = Fn.new { |br, n, l, sum, cnt|
    var b = br + 1
    while (b <= branches) {
        sum = sum + n
        if (sum > nMax) return
        if (l*2 >= sum && b >= branches) return
        if (b == br + 1) {
            c[br] = rooted[n] * cnt
        } else {
            var tmp = rooted[n] + BigInt.new(b - br - 1)
            c[br] = c[br] * tmp
            c[br] = c[br] / BigInt.new(b - br)
        }
        if (l*2 < sum) unrooted[sum] = unrooted[sum] + c[br]
        if (b < branches) rooted[sum] = rooted[sum] + c[br]
        var m = n - 1
        while (m > 0) {
            tree.call(b, m, l, sum, c[br])
            m = m - 1
        }
        b = b + 1
    }
}

var bicenter = Fn.new { |s|
    if (s%2 == 0) {
        var tmp = (rooted[(s/2).floor] + BigInt.one) * rooted[(s/2).floor]
        tmp = tmp >> 1
        unrooted[s] = unrooted[s] + tmp
    }
}

rooted[0] = BigInt.one
rooted[1] = BigInt.one
unrooted[0] = BigInt.one
unrooted[1] = BigInt.one
for (n in 1..nMax) {
    tree.call(0, n, n, 1, BigInt.one)
    bicenter.call(n)
    Fmt.print("$3d: $i", n, unrooted[n])
}
Output:

Abbreviated.

  1: 1
  2: 1
  3: 1
  4: 2
  5: 3
  6: 5
  7: 9
  8: 18
  9: 35
 10: 75
 11: 159
 12: 355
 13: 802
 14: 1858
 15: 4347
 16: 10359
 17: 24894
 18: 60523
 19: 148284
 20: 366319
 21: 910726
 22: 2278658
 23: 5731580
 24: 14490245
 25: 36797588
 26: 93839412
 27: 240215803
 28: 617105614
 29: 1590507121
 30: 4111846763
 31: 10660307791
 32: 27711253769
 33: 72214088660
 34: 188626236139
 35: 493782952902
...
249: 5814271898167303040368103945830220447130073898083466852225709084407144308593691069932064987528870826155297
250: 16206624309085062837751018464745815688226709117091506494175397665527493805947344857313038875654104100026504

zkl

Translation of: D
Translation of: Go

Uses GMP for big ints, mostly modified in place. Rather slow.

var BN=Import("zklBigNum");

const nMax=100, nBranches=4;
 
var rooted  =(nMax+1).pump(List.createLong(nMax+1).write,BN.fp(0)),
    unrooted=(nMax+1).pump(List.createLong(nMax+1).write,BN.fp(0));
rooted[0]=BN(1); rooted[1]=BN(1); unrooted[0]=BN(1); unrooted[1]=BN(1);
 
fcn tree(br,n,l,inSum,cnt){
   var c=(nBranches).pump(List().write,0);  // happens only once

   sum := inSum;
   foreach b in ([br + 1 .. nBranches]){
      sum += n;
      if (sum > nMax or (l * 2 >= sum and b >= nBranches)) return();
      if (b == br + 1) c[br] = rooted[n] * cnt; // -->BigInt
      else{
	 c[br].mul(rooted[n] + b - br - 1);
	 c[br].div(b - br);
      }
      if (l * 2 < sum) unrooted[sum].add(c[br]);
      if (b < nBranches) rooted[sum].add(c[br]);
      foreach m in ([n-1 .. 1,-1]) { tree(b, m, l, sum, c[br]); }
   }
}
 
fcn bicenter(s){
   if (s.isEven) unrooted[s].add(rooted[s / 2] * (rooted[s / 2] + 1) / 2);
}
 
foreach n in ([1 .. nMax]){
   tree(0, n, n, 1, BN(1));
   bicenter(n);
   println(n, ": ", unrooted[n]);
}
Output:
1: 1
2: 1
3: 1
4: 2
5: 3
6: 5
7: 9
8: 18
9: 35
10: 75
...
97: 286312976836850192359345859166390622180
98: 785684759853087702778573182234297830503
99: 2156596319845084996862701478402986311496
100: 5921072038125809849884993369103538010139