Faulhaber's formula: Difference between revisions
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=={{header|C}}==
{{trans|Modula-2}}
<
#include <stdio.h>
#include <stdlib.h>
Line 189:
return 0;
}</
{{out}}
<pre>0 : n
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=={{header|C sharp|C#}}==
{{trans|Java}}
<
namespace FaulhabersFormula {
Line 383:
}
}
}</
{{out}}
<pre>0 : n
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{{trans|D}}
Uses C++17
<
#include <numeric>
#include <sstream>
Line 568:
return 0;
}</
{{out}}
<pre>0 : n
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=={{header|D}}==
{{trans|Kotlin}}
<
import std.exception : enforce;
import std.format : formattedWrite;
Line 732:
faulhaber(i);
}
}</
{{out}}
<pre>0 : n
Line 746:
=={{header|EchoLisp}}==
<
(lib 'math) ;; for bernoulli numbers
(string-delimiter "")
Line 764:
(define (Faulcomp n p)
(printf "Σ(1..%d) n^%d = %d" n p (/ (poly n (Faulhaber p)) (1+ p) )))
</syntaxhighlight>
{{out}}
<pre>
Line 794:
=={{header|Factor}}==
<
math.functions regexp sequences ;
Line 812:
: poly>str ( seq -- str ) (poly>str) clean-up ;
10 [ dup faulhaber poly>str "%d: %s\n" printf ] each-integer</
{{out}}
<pre>
Line 826:
9: 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
</pre>
=={{header|FreeBASIC}}==
{{trans|C}}
<syntaxhighlight lang="vbnet">Type Fraction
As Integer num
As Integer den
End Type
Function Binomial(n As Integer, k As Integer) As Integer
If n < 0 Or k < 0 Then Print "Arguments must be non-negative integers": Exit Function
If n < k Then Print "The second argument cannot be more than the first.": Exit Function
If n = 0 Or k = 0 Then Return 1
Dim As Integer i, num, den
num = 1
For i = k + 1 To n
num *= i
Next i
den = 1
For i = 2 To n - k
den *= i
Next i
Return num / den
End Function
Function GCD(n As Integer, d As Integer) As Integer
Return Iif(d = 0, n, GCD(d, n Mod d))
End Function
Function makeFrac(n As Integer, d As Integer) As Fraction
Dim As Fraction result
Dim As Integer g
If d = 0 Then
result.num = 0
result.den = 0
Return result
End If
If n = 0 Then
d = 1
Elseif d < 0 Then
n = -n
d = -d
End If
g = Abs(gcd(n, d))
If g > 1 Then
n /= g
d /= g
End If
result.num = n
result.den = d
Return result
End Function
Function negateFrac(f As Fraction) As Fraction
Return makeFrac(-f.num, f.den)
End Function
Function subFrac(lhs As Fraction, rhs As Fraction) As Fraction
Return makeFrac(lhs.num * rhs.den - lhs.den * rhs.num, rhs.den * lhs.den)
End Function
Function multFrac(lhs As Fraction, rhs As Fraction) As Fraction
Return makeFrac(lhs.num * rhs.num, lhs.den * rhs.den)
End Function
Function equalFrac(lhs As Fraction, rhs As Fraction) As Integer
Return (lhs.num = rhs.num) And (lhs.den = rhs.den)
End Function
Function lessFrac(lhs As Fraction, rhs As Fraction) As Integer
Return (lhs.num * rhs.den) < (rhs.num * lhs.den)
End Function
Sub printFrac(f As Fraction)
Print Str(f.num);
If f.den <> 1 Then Print "/" & f.den;
End Sub
Function Bernoulli(n As Integer) As Fraction
If n < 0 Then Print "Argument must be non-negative": Exit Function
Dim As Fraction a(16)
Dim As Integer j, m
If (n < 0) Then
a(0).num = 0
a(0).den = 0
Return a(0)
End If
For m = 0 To n
a(m) = makeFrac(1, m + 1)
For j = m To 1 Step -1
a(j - 1) = multFrac(subFrac(a(j - 1), a(j)), makeFrac(j, 1))
Next j
Next m
If n <> 1 Then Return a(0)
Return negateFrac(a(0))
End Function
Sub Faulhaber(p As Integer)
Dim As Fraction coeff, q
Dim As Integer j, pwr, sign
Print p & " : ";
q = makeFrac(1, p + 1)
sign = -1
For j = 0 To p
sign = -1 * sign
coeff = multFrac(multFrac(multFrac(q, makeFrac(sign, 1)), makeFrac(Binomial(p + 1, j), 1)), Bernoulli(j))
If (equalFrac(coeff, makeFrac(0, 1))) Then Continue For
If j = 0 Then
If Not equalFrac(coeff, makeFrac(1, 1)) Then
If equalFrac(coeff, makeFrac(-1, 1)) Then
Print "-";
Else
printFrac(coeff)
End If
End If
Else
If equalFrac(coeff, makeFrac(1, 1)) Then
Print " + ";
Elseif equalFrac(coeff, makeFrac(-1, 1)) Then
Print " - ";
Elseif lessFrac(makeFrac(0, 1), coeff) Then
Print " + ";
printFrac(coeff)
Else
Print " - ";
printFrac(negateFrac(coeff))
End If
End If
pwr = p + 1 - j
Print Iif(pwr > 1, "n^" & pwr, "n");
Next j
Print
End Sub
For i As Integer = 0 To 9
Faulhaber(i)
Next i
Sleep</syntaxhighlight>
{{out}}
<pre>Same as C entry.</pre>
=={{header|Fōrmulæ}}==
{{FormulaeEntry|page=https://formulae.org/?script=examples/Faulhaber}}
'''Solution'''
The following function creates the Faulhaber's coefficients up to a given number of rows, according to the [http://www.ww.ingeniousmathstat.org/sites/default/files/Torabi-Dashti-CMJ-2011.pdf paper] of of Mohammad Torabi Dashti:
(This is exactly the as than the task [[Faulhaber%27s_triangle#F%C5%8Drmul%C3%A6|Faulhaber's triangle]])
[[File:Fōrmulæ - Faulhaber 01.png]]
The following function creates the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n:
Notes. The -1 index means the last element (-2 is the penultimate element, and so on). So it retrieves the last row of the triangle. |x| is the cardinality (number of elements) of x.
(This is exactly the as than the task [[Faulhaber%27s_triangle#F%C5%8Drmul%C3%A6|Faulhaber's triangle]])
This function can be used for both symbolic or numeric computation of the polynomial:
[[File:Fōrmulæ - Faulhaber 06.png]]
To generate the first 10 closed-form expressions, starting with p = 0:
[[File:Fōrmulæ - Faulhaber 07.png]]
[[File:Fōrmulæ - Faulhaber 08.png]]
=={{header|GAP}}==
Straightforward implementation using GAP polynomials, and two different formulas: one based on Stirling numbers of the second kind (sum1, see Python implementation below in this page), and the usual Faulhaber formula (sum2). No optimization is made (one could compute Stirling numbers row by row, or the product in sum1 may be kept from one call to the other). Notice the Bernoulli term in the first formula is here only to correct the value of sum1(0), which is off by one because sum1 computes sums from 0 to n.
<
sum1 := p -> Sum([0 .. p], k -> Stirling2(p, k) * Product([0 .. k], j -> n + 1 - j) / (k + 1)) + 2 * Bernoulli(2 * p + 1);
sum2 := p -> Sum([0 .. p], j -> (-1)^j * Binomial(p + 1, j) * Bernoulli(j) * n^(p + 1 - j)) / (p + 1);
Line 856 ⟶ 1,026:
1/8*n^8+1/2*n^7+7/12*n^6-7/24*n^4+1/12*n^2
1/9*n^9+1/2*n^8+2/3*n^7-7/15*n^5+2/9*n^3-1/30*n
1/10*n^10+1/2*n^9+3/4*n^8-7/10*n^6+1/2*n^4-3/20*n^2</
=={{header|Go}}==
<
import (
Line 918 ⟶ 1,088:
fmt.Println()
}
}</
{{out}}
<pre>
Line 935 ⟶ 1,105:
=={{header|Groovy}}==
{{trans|Java}}
<
class FaulhabersFormula {
Line 1,076 ⟶ 1,246:
}
}
}</
{{out}}
<pre>0 : n
Line 1,091 ⟶ 1,261:
=={{header|Haskell}}==
====Bernouilli polynomials====
<
import Data.List (intercalate, transpose)
import Data.Bifunctor (bimap)
Line 1,193 ⟶ 1,363:
unsignedLength xs =
let l = length xs
in bool (bool l (l - 1) ('-' == head xs)) 0 (0 == l)</
{{Out}}
<pre>0 -> n
Line 1,210 ⟶ 1,380:
Implementation:
<
+/,(<:*(_1^[)*!*(y^~1+[)%1+])"0/~i.1x+y
)
Line 1,218 ⟶ 1,388:
Faul=:adverb define
(0,|.(%m+1x) * (_1x&^ * !&(m+1) * Bfirst) i.1+m)&p.
)</
Task example:
<
0 1x&p.
1 Faul
Line 1,241 ⟶ 1,411:
0 _1r30 0 2r9 0 _7r15 0 2r3 1r2 1r9&p.
9 Faul
0 0 _3r20 0 1r2 0 _7r10 0 3r4 1r2 1r10&p.</
Double checking our work:
<
9 Faul i.5
0 1 513 20196 282340
Line 1,253 ⟶ 1,423:
0 1 5 14 30
2 Fcheck i.5
0 1 5 14 30</
=={{header|Java}}==
{{trans|Kotlin}}
{{works with|Java|8}}
<
import java.util.stream.IntStream;
Line 1,399 ⟶ 1,569:
}
}
}</
{{out}}
<pre>0 : n
Line 1,411 ⟶ 1,581:
8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n
9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2</pre>
=={{header|jq}}==
{{works with|jq}}
'''Works with gojq, the Go implementation of jq, and with fq'''
The following assumes the following rational arithmetic functions
as provided by the jq "rational" module at [[Arithmetic/Rational#jq]]:
* r/2 (constructor)
* requal/2 (equality)
* rlessthan/1
* rminus/1
* rminus/2 (subtraction)
* rmult/0 and rmult/2 (multiplication)
In addition, the pretty-printing function defined here (rpp) assumes
rationals are JSON objects of the form {n,d}
<syntaxhighlight lang=jq>
include "rational" {search: "."}; # see [[Arithmetic/Rational#jq]]:
# Preliminaries
# for gojq
def idivide($j):
. as $i
| ($i % $j) as $mod
| ($i - $mod) / $j ;
# use idivide for precision
def binomial(n; k):
if k > n / 2 then binomial(n; n-k)
else reduce range(1; k+1) as $i (1; . * (n - $i + 1) | idivide($i))
end;
# pretty print a Rational assumed to have the {n,d} form
def rpp:
if .n == 0 then "0"
elif .d == 1 then .n | tostring
else "\(.n)/\(.d)"
end;
# The following definition reflects the "modern view" that B(1) is 1 // 2
def bernoulli:
if type != "number" or . < 0 then "bernoulli must be given a non-negative number vs \(.)" | error
else . as $n
| reduce range(0; $n+1) as $i ([];
.[$i] = r(1; $i + 1)
| reduce range($i; 0; -1) as $j (.;
.[$j-1] = rmult($j; rminus(.[$j-1]; .[$j])) ) )
| .[0] # the modern view
end;
# The task
def faulhaber($p):
# The traditional view:
def bernouilli($n):
$n | bernouilli | if $n==1 then rminus else . end;
r(1; $p+1) as $q
| { sign: -1 }
| reduce range(0; 1+$p) as $j (.;
.sign *= -1
| r(binomial($p+1; $j); 1) as $b
| ([$q, .sign, $b, ($j|bernoulli)] | rmult) as $coeff
| if requal($coeff; r(0;1))|not
then .emit +=
(if $j == 0
then (if requal($coeff; r( 1;1)) then ""
elif requal($coeff; r(-1;1)) then "-"
else "\($coeff|rpp) " end)
else (if requal($coeff; r(1;1)) then " + "
elif requal($coeff; r(-1;1)) then " - "
elif r(0;1)|rlessthan($coeff) then " + \($coeff|rpp) "
else " - \($coeff|rminus|rpp) "
end)
end)
| ($p + 1 - $j) as $pwr
| .emit += (if 1 < $pwr then "n^\($pwr)" else "n" end)
else .
end )
| .emit ;
range(0;10) | "\(.) : \(faulhaber(.))"
</syntaxhighlight>
{{output}}
<pre>
0 : n
1 : 1/2 n^2 - 1/2 n
2 : 1/3 n^3 - 1/2 n^2 + 1/6 n
3 : 1/4 n^4 - 1/2 n^3 + 1/4 n^2
4 : 1/5 n^5 - 1/2 n^4 + 1/3 n^3 - 1/30 n
5 : 1/6 n^6 - 1/2 n^5 + 5/12 n^4 - 1/12 n^2
6 : 1/7 n^7 - 1/2 n^6 + 1/2 n^5 - 1/6 n^3 + 1/42 n
7 : 1/8 n^8 - 1/2 n^7 + 7/12 n^6 - 7/24 n^4 + 1/12 n^2
8 : 1/9 n^9 - 1/2 n^8 + 2/3 n^7 - 7/15 n^5 + 2/9 n^3 - 1/30 n
9 : 1/10 n^10 - 1/2 n^9 + 3/4 n^8 - 7/10 n^6 + 1/2 n^4 - 3/20 n^2
</pre>
=={{header|Julia}}==
Line 1,416 ⟶ 1,683:
'''Module''':
<
function bernoulli(n::Integer)
Line 1,456 ⟶ 1,723:
end
end # module Faulhaber</
'''Main''':
<syntaxhighlight lang
{{out}}
Line 1,475 ⟶ 1,742:
=={{header|Kotlin}}==
As Kotlin doesn't have support for rational numbers built in, a cut-down version of the Frac class in the Arithmetic/Rational task has been used in order to express the polynomial coefficients as fractions.
<
fun gcd(a: Long, b: Long): Long = if (b == 0L) a else gcd(b, a % b)
Line 1,593 ⟶ 1,860:
fun main(args: Array<String>) {
for (i in 0..9) faulhaber(i)
}</
{{out}}
Line 1,611 ⟶ 1,878:
=={{header|Lua}}==
{{trans|C}}
<
if n<0 or k<0 or n<k then return -1 end
if n==0 or k==0 then return 1 end
Line 1,756 ⟶ 2,023:
for i=0,9 do
faulhaber(i)
end</
{{out}}
<pre>0 : n
Line 1,770 ⟶ 2,037:
=={{header|Mathematica}} / {{header|Wolfram Language}}==
<
Faulhaber[n_, 0] := n
Faulhaber[n_, p_] := n^(p + 1)/(p + 1) + 1/2 n^p + Sum[BernoulliB[k]/k! p!/(p - k + 1)! n^(p - k + 1), {k, 2, p}]
Table[{p, Faulhaber[n, p]}, {p, 0, 9}] // Grid</
{{out}}
<pre>0 n
Line 1,787 ⟶ 2,054:
=={{header|Maxima}}==
<
sum2(p):=sum((-1)^j*binomial(p+1,j)*bern(j)*n^(p-j+1),j,0,p)/(p+1)$
Line 1,793 ⟶ 2,060:
[0,0,0,0,0,0,0,0,0,0]
for p from 0 thru 9 do print(expand(sum2(p)));</
{{out}}
<pre>
Line 1,810 ⟶ 2,077:
=={{header|Modula-2}}==
{{trans|C#}}
<
FROM EXCEPTIONS IMPORT AllocateSource,ExceptionSource,GetMessage,RAISE;
FROM FormatString IMPORT FormatString;
Line 1,991 ⟶ 2,258:
END;
ReadChar
END Faulhaber.</
{{out}}
<pre>0 : n
Line 2,006 ⟶ 2,273:
=={{header|Nim}}==
{{trans|Kotlin}}
<
type
Line 2,079 ⟶ 2,346:
for n in 0..9:
echo n, ": ", faulhaber(n)</
{{out}}
Line 2,101 ⟶ 2,368:
It's not worth using Bernoulli numbers in PARI/GP, because too much cleaning if you are avoiding "dirty" (but correct) result.<br>
Note: Find ssubstr() function here on RC.
<
\\ Using "Faulhaber's" formula based on Bernoulli numbers. aev 2/7/17
\\ In str string replace all occurrences of the search string ssrch with the replacement string srepl. aev 3/8/16
Line 2,133 ⟶ 2,400:
{\\ Testing:
for(i=0,9, Faulhaber2(i))}
</syntaxhighlight>
{{Output}}
<pre>
Line 2,151 ⟶ 2,418:
This version is using the sums of pth powers formula from [[wp:Bernoulli_polynomials| Bernoulli polynomials]].
It has small, simple and clear code, and produces instant result.
<
\\ Using a formula based on Bernoulli polynomials. aev 2/5/17
Faulhaber1(m)={
Line 2,162 ⟶ 2,429:
{\\ Testing:
for(i=0,9, Faulhaber1(i))}
</syntaxhighlight>
{{Output}}
<pre>
Line 2,177 ⟶ 2,444:
> ##
*** last result computed in 0 ms
</pre>
=={{header|Pascal}}==
A console program that runs under Lazarus or Delphi. Does not make use of Bernoulli numbers.
<syntaxhighlight lang="pascal">
program Faulhaber;
{$IFDEF FPC} // Lazarus
{$MODE Delphi} // ensure Lazarus accepts Delphi-style code
{$ASSERTIONS+} // by default, Lazarus does not compile 'Assert' statements
{$ELSE} // Delphi
{$APPTYPE CONSOLE}
{$ENDIF}
uses SysUtils;
type TRational = record
Num, Den : integer; // where Den > 0 and Num, Den are coprime
end;
const
ZERO : TRational = ( Num: 0; Den : 1);
HALF : TRational = ( Num: 1; Den : 2);
// Construct rational a/b, assuming b > 0.
function Rational( const a, b : integer) : TRational;
var
t, x, y : integer;
begin
if b <= 0 then raise SysUtils.Exception.Create( 'Denominator must be > 0');
// Find HCF of a and b (Euclid's algorithm) and cancel it out.
x := Abs(a);
y := b;
while y <> 0 do begin
t := x mod y;
x := y;
y := t;
end;
result.Num := a div x;
result.Den := b div x
end;
function Prod( r, s : TRational) : TRational; // result := r*s
begin
result := Rational( r.Num*s.Num, r.Den*s.Den);
end;
procedure DecRat( var r : TRational;
const s : TRational); // r := r - s
begin
r := Rational( r.Num*s.Den - s.Num*r.Den, r.Den * s.Den);
end;
// Write a term such as ' - (7/10)n^6' to the console.
procedure WriteTerm( coeff : TRational;
index : integer;
printPlus : boolean);
begin
if Coeff.Num = 0 then exit;
with coeff do begin
if Num < 0 then Write(' - ')
else if printPlus then Write(' + ');
// Put brackets round a fractional coefficient
if (Den > 1) then Write('(');
// If coefficient is 1, don't write it
if (Den > 1) or (Abs(Num) > 1) then Write( Abs(Num));
// Write denominator if it's not 1
if (Den > 1) then Write('/', Den, ')');
end;
Write('n');
if index > 1 then Write('^', index);
end;
{-------------------------------------------------------------------------------
Main routine. Calculation of Faulhaber polynomials
F_p(n) = 1^p + 2^p + ... + n^p, p = 0, 1, ..., p_max
}
var
p_max : integer;
c : array of array of TRational;
i, j, p : integer;
coeff_of_n : TRational;
begin
// User types program name, optionally followed by maximum power p (defaults to 9)
if ParamCount = 0 then p_max := 9
else p_max := SysUtils.StrToInt( ParamStr(1));
// c[p, i] is coefficient of n^i in the polynomial F_p(n).
// Initialize all coefficients to 0.
SetLength( c, p_max + 1, p_max + 2);
for i := 0 to p_max do
for j := 0 to p_max + 1 do
c[i, j] := ZERO;
c[0, 1] := Rational(1, 1); // F_0(n) = n, special case
for p := 1 to p_max do begin
// Initialize calculation of coefficient of n, needed if p is even.
// If p is odd, still calculate it as a check on the working (should be 0).
// Calculation uses the fact that F_p(1) = 1.
coeff_of_n := Rational(1, 1);
c[p, p+1] := Rational(1, p + 1);
DecRat( coeff_of_n, c[p, p + 1]);
c[p, p] := HALF;
DecRat( coeff_of_n, c[p, p]);
i := p - 1;
while (i >= 2) do begin
c[p, i] := Prod( Rational(p, i), c[p - 1, i - 1]);
DecRat( coeff_of_n, c[p, i]);
dec(i, 2);
end;
if i = 1 then // p is even
c[p, 1] := coeff_of_n // = the Bernoulli number B_p
else // p is odd
Assert( coeff_of_n.Num = 0); // just checking
end; // for p
// Print the result
for p := 0 to p_max do begin
Write( 'F_', p, '(n) = ');
for j := p + 1 downto 1 do WriteTerm( c[p, j], j, j <= p);
WriteLn;
end;
end.
</syntaxhighlight>
{{out}}
<pre>
F_0(n) = n
F_1(n) = (1/2)n^2 + (1/2)n
F_2(n) = (1/3)n^3 + (1/2)n^2 + (1/6)n
F_3(n) = (1/4)n^4 + (1/2)n^3 + (1/4)n^2
F_4(n) = (1/5)n^5 + (1/2)n^4 + (1/3)n^3 - (1/30)n
F_5(n) = (1/6)n^6 + (1/2)n^5 + (5/12)n^4 - (1/12)n^2
F_6(n) = (1/7)n^7 + (1/2)n^6 + (1/2)n^5 - (1/6)n^3 + (1/42)n
F_7(n) = (1/8)n^8 + (1/2)n^7 + (7/12)n^6 - (7/24)n^4 + (1/12)n^2
F_8(n) = (1/9)n^9 + (1/2)n^8 + (2/3)n^7 - (7/15)n^5 + (2/9)n^3 - (1/30)n
F_9(n) = (1/10)n^10 + (1/2)n^9 + (3/4)n^8 - (7/10)n^6 + (1/2)n^4 - (3/20)n^2
</pre>
=={{header|Perl}}==
<
use Math::Algebra::Symbols;
Line 2,222 ⟶ 2,626:
foreach my $i (0 .. 9) {
say "$i: ", faulhaber_s_formula($i);
}</
{{out}}
<pre>
Line 2,239 ⟶ 2,643:
=={{header|Phix}}==
{{trans|C#}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #000000;">builtins</span><span style="color: #0000FF;">\</span><span style="color: #000000;">pfrac</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span> <span style="color: #000080;font-style:italic;">-- (0.8.0+)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">bernoulli</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">m</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_mul</span><span style="color: #0000FF;">({</span><span style="color: #000000;">j</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span><span style="color: #000000;">frac_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">frac_uminus</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">binomial</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">k</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">k</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">num</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">denom</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">num</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">i</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">denom</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">i</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">num</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">denom</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">faulhaber</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%d : "</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">frac</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">p</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">frac</span> <span style="color: #000000;">bj</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernoulli</span><span style="color: #0000FF;">(</span><span style="color: #000000;">j</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">frac_ne</span><span style="color: #0000FF;">(</span><span style="color: #000000;">bj</span><span style="color: #0000FF;">,</span><span style="color: #000000;">frac_zero</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">frac</span> <span style="color: #000000;">coeff</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_mul</span><span style="color: #0000FF;">({</span><span style="color: #000000;">binomial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j</span><span style="color: #0000FF;">),</span><span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span><span style="color: #000000;">bj</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_sprint</span><span style="color: #0000FF;">(</span><span style="color: #000000;">coeff</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">=</span><span style="color: #008000;">"1"</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span
<span style="color: #008080;">if</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'-'</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">" - "</span>
<span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">0</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">" + "</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">&</span><span style="color: #008000;">"n"</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">pwr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">j</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">pwr</span><span style="color: #0000FF;">></span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"^%d"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">pwr</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">res</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">9</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">faulhaber</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Line 2,325 ⟶ 2,732:
<
def nextu(a):
Line 2,387 ⟶ 2,794:
for i, p in enumerate(sumpol(10)):
print(i, ":", polstr(p))</
{{out}}
Line 2,405 ⟶ 2,812:
Racket will simplify rational numbers; if this code simplifies the expressions too much for your tastes (e.g. you like <code>1/1 * (n)</code>) then tweak the simplify... clauses to taste.
<
(require racket/match
Line 2,461 ⟶ 2,868:
(for ((p (in-range 0 (add1 9))))
(printf "f(~a) = ~a~%" p (expression->infix-string (faulhaber p))))
</syntaxhighlight>
{{out}}
Line 2,480 ⟶ 2,887:
(formerly Perl 6)
{{works with|Rakudo|2018.04.01}}
<syntaxhighlight lang="raku"
return 1/2 if $n == 1;
Line 2,518 ⟶ 2,925:
for 0..9 -> $p {
say "f($p) = ", faulhaber_s_formula($p);
}</
{{out}}
<pre>
Line 2,531 ⟶ 2,938:
f(8) = 1/9 * (n^9 + (9/2)*n^8 + (6/1)*n^7 + (-21/5)*n^5 + (2/1)*n^3 + (-3/10)*n)
f(9) = 1/10 * (n^10 + (5/1)*n^9 + (15/2)*n^8 + (-7/1)*n^6 + (5/1)*n^4 + (-3/2)*n^2)
</pre>
=={{header|RPL}}==
{{works with|HP|49}}
≪ → p
≪ 0
p 0 '''FOR''' m
p 1 + m 1 + COMB
p m - IBERNOULLI
'''IF''' LASTARG 1 == '''THEN''' NEG '''END'''
* EVAL + 'n' *
-1 '''STEP'''
p 1 + / EXPAN
≫ ≫ '<span style="color:blue>FAULH</span>' STO
≪ P <span style="color:blue>FAULH</span> ≫ 'P' 0 9 1 SEQ
{{out}}
<pre>
1: {'n' '1/2n^2+1/2n' '1/3n^3+1/2n^2+1/6n' '1/4n^4+1/2n^3+1/4n^2' '1/5n^5+1/2n^4+1/3n^3-1/30n' '1/6n^6+1/2n^5+5/12n^4-1/12n^2' '1/7n^7+1/2n^6+1/2n^5-1/6n^3+1/42n' '1/8n^8+1/2n^7+7/12n^6-7/24n^4+1/12n^2' '1/9n^9+1/2n^8+2/3n^7-7/15n^5+2/9n^3-1/30n' '1/10n^10+1/2n^9+3/4n^8-7/10n^6+1/2n^4-3/20n^2'}
</pre>
=={{header|Ruby}}==
{{trans|C}}
<
if n < 0 or k < 0 or n < k then
return -1
Line 2,622 ⟶ 3,048:
end
main()</
{{out}}
<pre>0 : n
Line 2,637 ⟶ 3,063:
=={{header|Scala}}==
{{trans|Java}}
<
abstract class Frac extends Comparable[Frac] {
Line 2,807 ⟶ 3,233:
}
}
}</
{{out}}
<pre>0 : n
Line 2,821 ⟶ 3,247:
=={{header|Sidef}}==
<syntaxhighlight lang="ruby">func faulhaber_formula(p) {
(p+1).of { |j|
Poly
}.sum / (p+1)
}
for p in (^10) {
printf("%2d: %s\n", p, faulhaber_formula(p))
}</
{{out}}
<pre>
0:
1: 1/2
2: 1/3
3: 1/4
4: 1/5
5: 1/6
6: 1/7
7: 1/8
8: 1/9
9: 1/10
</pre>
=={{header|Visual Basic .NET}}==
{{trans|C#}}
<
Function Gcd(a As Long, b As Long)
If b = 0 Then
Line 3,022 ⟶ 3,434:
Next
End Sub
End Module</
{{out}}
<pre>0 : n
Line 3,039 ⟶ 3,451:
{{libheader|Wren-math}}
{{libheader|Wren-rat}}
<
import "./rat" for Rat
var bernoulli = Fn.new { |n|
Line 3,091 ⟶ 3,503:
}
for (i in 0..9) faulhaber.call(i)</
{{out}}
Line 3,110 ⟶ 3,522:
{{libheader|GMP}} GNU Multiple Precision Arithmetic Library
Uses code from the Bernoulli numbers task (copied here).
<
fcn faulhaberFormula(p){ //-->(Rational,Rational...)
[p..0,-1].pump(List(),'wrap(k){ B(k)*BN(p+1).binomial(k) })
.apply('*(Rational(1,p+1)))
}</
<
println("F(%d) --> %s".fmt(p,polyRatString(faulhaberFormula(p))))
}</
<
fcn init(_a,_b){ var a=_a, b=_b; normalize(); }
fcn toString{
Line 3,142 ⟶ 3,554:
}
fcn __opDiv(n){ self(a*n.b,b*n.a) } // Rat / Rat
}</
<
var A=List.createLong(100,0); // aka static aka not thread safe
foreach m in (N+1){
Line 3,159 ⟶ 3,571:
if(str[0]=="+") str[1,*]; // leave leading space
else String("-",str[2,*]);
}</
{{out}}
<pre>
|