Binomial transform: Difference between revisions

From Rosetta Code
Content added Content deleted
(Add Python)
(Added Go)
Line 55: Line 55:
;* [[oeis:A034943|OEIS:A034943 - Binomial transform of Padovan sequence]]
;* [[oeis:A034943|OEIS:A034943 - Binomial transform of Padovan sequence]]
;* [[oeis:A144413|OEIS:A144413 - a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4) (Inverse binomial transform of Padovan sequence)]]
;* [[oeis:A144413|OEIS:A144413 - a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4) (Inverse binomial transform of Padovan sequence)]]

=={{header|Go}}==
<syntaxhighlight lang="go">package main

import "fmt"

func factorial(n int) uint64 {
if n > 20 {
return 0 // too big for uint64
}
if n < 2 {
return 1
}
fact := uint64(1)
i := 2
for i <= n {
fact *= uint64(i)
i++
}
return fact
}

func binomial(n, k int) uint64 {
return factorial(n) / factorial(n-k) / factorial(k)
}

func btForward(a []int64) []int64 {
c := len(a)
b := make([]int64, c)
for n := 0; n < c; n++ {
b[n] = int64(0)
for k := 0; k <= n; k++ {
b[n] += int64(binomial(n, k)) * a[k]
}
}
return b
}

func btInverse(b []int64) []int64 {
c := len(b)
a := make([]int64, c)
for n := 0; n < c; n++ {
a[n] = int64(0)
for k := 0; k <= n; k++ {
sign := int64(-1)
if (n-k)&1 == 0 {
sign = 1
}
a[n] += int64(binomial(n, k)) * b[k] * sign
}
}
return a
}

func main() {
seqs := [][]int64{
{1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190},
{0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0},
{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181},
{1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37},
}

names := []string{
"Catalan number sequence:",
"Prime flip-flop sequence:",
"Fibonacci number sequence:",
"Padovan number sequence:",
}

for i, seq := range seqs {
fmt.Println(names[i])
fmt.Println(seq)
fmt.Println("Forward binomial transform:")
fwd := btForward(seq)
fmt.Println(fwd)
fmt.Println("Inverse binomial transform:")
fmt.Println(btInverse(seq))
fmt.Println("Round trip:")
fmt.Println(btInverse(fwd))
if i < len(seqs)-1 {
fmt.Println()
}
}
}</syntaxhighlight>

{{out}}
<pre>
Catalan number sequence:
[1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190]
Forward binomial transform:
[1 2 5 15 51 188 731 2950 12235 51822 223191 974427 4302645 19181100 86211885 390248055 1777495635 8140539950 37463689775 173164232965]
Inverse binomial transform:
[1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595]
Round trip:
[1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190]

Prime flip-flop sequence:
[0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0]
Forward binomial transform:
[0 1 3 6 11 20 37 70 134 255 476 869 1564 2821 5201 9948 19793 40562 84271 174952]
Inverse binomial transform:
[0 1 -1 0 3 -10 25 -56 118 -237 456 -847 1540 -2795 5173 -9918 19761 -40528 84235 -174914]
Round trip:
[0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0]

Fibonacci number sequence:
[0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181]
Forward binomial transform:
[0 1 3 8 21 55 144 377 987 2584 6765 17711 46368 121393 317811 832040 2178309 5702887 14930352 39088169]
Inverse binomial transform:
[0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144 233 -377 610 -987 1597 -2584 4181]
Round trip:
[0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181]

Padovan number sequence:
[1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37]
Forward binomial transform:
[1 1 1 2 5 12 28 65 151 351 816 1897 4410 10252 23833 55405 128801 299426 696081 1618192]
Inverse binomial transform:
[1 -1 1 0 -3 10 -24 49 -89 145 -208 245 -174 -176 1121 -3185 7137 -13920 24301 -37926]
Round trip:
[1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37]
</pre>


=={{header|Python}}==
=={{header|Python}}==

Revision as of 10:33, 6 May 2023

Binomial transform is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The binomial transform is a bijective sequence transform based on convolution with binomial coefficients.

It may be thought of as an nth forward difference with odd differences carrying a negative sign.

There are two common variants of binomial transforms, one of which is self-inverting; reapplying the transform to a transformed sequence returns the original sequence, and one which has separate "forward" and "inverse" transform operations.

The two variants only differ in placement and quantity of signs. The variant standardized on by OEIS, with the 'forward' and 'inverse' complementary operations, will be used here.


In this variant, to transform the sequence a to sequence b and back:

the forward binomial transform is defined as:

and the inverse binomial transform is defined as:

where is the binomial operator 'n choose k'.


Task
  • Implement both a forward, and inverse binomial transform routine.
  • Use those routines to compute the forward binomial transform, the inverse binomial transform, and the inverse of the forward transform (should return original sequence) of a few representative sequences.
  • Show at least the first 15 values in each sequence.
You may generate the sequences, or may choose to just hard code the values.
Use the following sequences for testing:
Catalan numbers: 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440
Prime flip flop sequence: (for an: 1 if prime, 0 otherwise): 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0
Fibonacci sequence: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377
Padovan sequence: (starting with values 1,0,0): 1 0 0 1 0 1 1 1 2 2 3 4 5 7 9


See also
first sequence
second sequence
third sequence
fourth sequence

Go

package main

import "fmt"

func factorial(n int) uint64 {
    if n > 20 {
        return 0 // too big for uint64
    }
    if n < 2 {
        return 1
    }
    fact := uint64(1)
    i := 2
    for i <= n {
        fact *= uint64(i)
        i++
    }
    return fact
}

func binomial(n, k int) uint64 {
    return factorial(n) / factorial(n-k) / factorial(k)
}

func btForward(a []int64) []int64 {
    c := len(a)
    b := make([]int64, c)
    for n := 0; n < c; n++ {
        b[n] = int64(0)
        for k := 0; k <= n; k++ {
            b[n] += int64(binomial(n, k)) * a[k]
        }
    }
    return b
}

func btInverse(b []int64) []int64 {
    c := len(b)
    a := make([]int64, c)
    for n := 0; n < c; n++ {
        a[n] = int64(0)
        for k := 0; k <= n; k++ {
            sign := int64(-1)
            if (n-k)&1 == 0 {
                sign = 1
            }
            a[n] += int64(binomial(n, k)) * b[k] * sign
        }
    }
    return a
}

func main() {
    seqs := [][]int64{
        {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190},
        {0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0},
        {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181},
        {1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37},
    }

    names := []string{
        "Catalan number sequence:",
        "Prime flip-flop sequence:",
        "Fibonacci number sequence:",
        "Padovan number sequence:",
    }

    for i, seq := range seqs {
        fmt.Println(names[i])
        fmt.Println(seq)
        fmt.Println("Forward binomial transform:")
        fwd := btForward(seq)
        fmt.Println(fwd)
        fmt.Println("Inverse binomial transform:")
        fmt.Println(btInverse(seq))
        fmt.Println("Round trip:")
        fmt.Println(btInverse(fwd))
        if i < len(seqs)-1 {
            fmt.Println()
        }
    }
}
Output:
Catalan number sequence:
[1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190]
Forward binomial transform:
[1 2 5 15 51 188 731 2950 12235 51822 223191 974427 4302645 19181100 86211885 390248055 1777495635 8140539950 37463689775 173164232965]
Inverse binomial transform:
[1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595]
Round trip:
[1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190]

Prime flip-flop sequence:
[0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0]
Forward binomial transform:
[0 1 3 6 11 20 37 70 134 255 476 869 1564 2821 5201 9948 19793 40562 84271 174952]
Inverse binomial transform:
[0 1 -1 0 3 -10 25 -56 118 -237 456 -847 1540 -2795 5173 -9918 19761 -40528 84235 -174914]
Round trip:
[0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0]

Fibonacci number sequence:
[0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181]
Forward binomial transform:
[0 1 3 8 21 55 144 377 987 2584 6765 17711 46368 121393 317811 832040 2178309 5702887 14930352 39088169]
Inverse binomial transform:
[0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144 233 -377 610 -987 1597 -2584 4181]
Round trip:
[0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181]

Padovan number sequence:
[1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37]
Forward binomial transform:
[1 1 1 2 5 12 28 65 151 351 816 1897 4410 10252 23833 55405 128801 299426 696081 1618192]
Inverse binomial transform:
[1 -1 1 0 -3 10 -24 49 -89 145 -208 245 -174 -176 1121 -3185 7137 -13920 24301 -37926]
Round trip:
[1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37]

Python

from math import factorial
from math import pow

from typing import Iterable
from typing import Sequence


def binomial(n: int, k: int) -> int:
    return factorial(n) // (factorial(n - k) * factorial(k))


def binomial_transform(seq: Sequence) -> Iterable[int]:
    for n in range(len(seq)):
        yield sum(binomial(n, k) * seq[k] for k in range(n + 1))


def inverse_binomial_transform(seq: Sequence) -> Iterable[int]:
    for n in range(len(seq)):
        yield int(sum(pow(-1, n - k) * binomial(n, k) * seq[k] for k in range(n + 1)))


test_sequences = {
    "Catalan number sequence": [
        1,
        1,
        2,
        5,
        14,
        42,
        132,
        429,
        1430,
        4862,
        16796,
        58786,
        208012,
        742900,
        2674440,
        9694845,
        35357670,
        129644790,
        477638700,
        1767263190,
    ],
    "Prime flip-flop sequence": [
        0,
        1,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
    ],
    "Fibonacci number sequence": [
        0,
        1,
        1,
        2,
        3,
        5,
        8,
        13,
        21,
        34,
        55,
        89,
        144,
        233,
        377,
        610,
        987,
        1597,
        2584,
        4181,
    ],
    "Padovan number sequence": [
        1,
        0,
        0,
        1,
        0,
        1,
        1,
        1,
        2,
        2,
        3,
        4,
        5,
        7,
        9,
        12,
        16,
        21,
        28,
        37,
    ],
}

if __name__ == "__main__":
    import pprint

    for desc, seq in test_sequences.items():
        print(desc + ":")
        pprint.pprint(seq, compact=True, indent=2)
        print("Forward binomial transform:")
        pprint.pprint(list(binomial_transform(seq)), compact=True, indent=2)
        print("Inverse binomial transform:")
        pprint.pprint(list(inverse_binomial_transform(seq)), compact=True, indent=2)
        print("Round trip:")
        pprint.pprint(
            list(inverse_binomial_transform(list(binomial_transform(seq)))),
            compact=True,
            indent=2,
        )
        print()
Output:
Catalan number sequence:
[ 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900,
  2674440, 9694845, 35357670, 129644790, 477638700, 1767263190]
Forward binomial transform:
[ 1, 2, 5, 15, 51, 188, 731, 2950, 12235, 51822, 223191, 974427, 4302645,
  19181100, 86211885, 390248055, 1777495635, 8140539950, 37463689775,
  173164232965]
Inverse binomial transform:
[ 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097,
  227475, 625992, 1730787, 4805595]
Round trip:
[ 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900,
  2674440, 9694845, 35357670, 129644790, 477638700, 1767263190]

Prime flip-flop sequence:
[0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0]
Forward binomial transform:
[ 0, 1, 3, 6, 11, 20, 37, 70, 134, 255, 476, 869, 1564, 2821, 5201, 9948, 19793,
  40562, 84271, 174952]
Inverse binomial transform:
[ 0, 1, -1, 0, 3, -10, 25, -56, 118, -237, 456, -847, 1540, -2795, 5173, -9918,
  19761, -40528, 84235, -174914]
Round trip:
[0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0]

Fibonacci number sequence:
[ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,
  4181]
Forward binomial transform:
[ 0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711, 46368, 121393, 317811,
  832040, 2178309, 5702887, 14930352, 39088169]
Inverse binomial transform:
[ 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, 233, -377, 610, -987,
  1597, -2584, 4181]
Round trip:
[ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,
  4181]

Padovan number sequence:
[1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37]
Forward binomial transform:
[ 1, 1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405,
  128801, 299426, 696081, 1618192]
Inverse binomial transform:
[ 1, -1, 1, 0, -3, 10, -24, 49, -89, 145, -208, 245, -174, -176, 1121, -3185,
  7137, -13920, 24301, -37926]
Round trip:
[1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37]

Raku

Generates the sequences on the fly. 

sub binomial { [×] ($^n … 0) Z/ 1 .. $^p }

sub binomial-transform (*@seq) {
    @seq.keys.map: -> \n { sum (0..n).map: -> \k { binomial(n,k) × @seq[k] } }
}

sub inverse-binomial-transform (*@seq) {
    @seq.keys.map: -> \n { sum (0..n).map: -> \k { exp(n - k, -1) × binomial(n,k) × @seq[k] } }
}

my $upto = 20;

for 'Catalan number',   (1, { [+] @_ Z× @_.reverse }…*),
    'Prime flip-flop',  (1..*).map({.is-prime ?? 1 !! 0}),
    'Fibonacci number', (0,1,1,*+*…*),
    'Padovan number',   (1,0,0, -> $c,$b,$ {$b+$c}…*)
  -> $name, @seq {
    say qq:to/BIN/;
    $name sequence:
    {@seq[^$upto]}
    Forward binomial transform:
    {binomial-transform(@seq)[^$upto]}
    Inverse binomial transform:
    {inverse-binomial-transform(@seq)[^$upto]}
    Round trip:
    {inverse-binomial-transform(binomial-transform(@seq))[^$upto]}
    BIN
}
Output:
Catalan number sequence:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190
Forward binomial transform:
1 2 5 15 51 188 731 2950 12235 51822 223191 974427 4302645 19181100 86211885 390248055 1777495635 8140539950 37463689775 173164232965
Inverse binomial transform:
1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595
Round trip:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190

Prime flip-flop sequence:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
Forward binomial transform:
0 1 3 6 11 20 37 70 134 255 476 869 1564 2821 5201 9948 19793 40562 84271 174952
Inverse binomial transform:
0 1 -1 0 3 -10 25 -56 118 -237 456 -847 1540 -2795 5173 -9918 19761 -40528 84235 -174914
Round trip:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0

Fibonacci number sequence:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
Forward binomial transform:
0 1 3 8 21 55 144 377 987 2584 6765 17711 46368 121393 317811 832040 2178309 5702887 14930352 39088169
Inverse binomial transform:
0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144 233 -377 610 -987 1597 -2584 4181
Round trip:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181

Padovan number sequence:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37
Forward binomial transform:
1 1 1 2 5 12 28 65 151 351 816 1897 4410 10252 23833 55405 128801 299426 696081 1618192
Inverse binomial transform:
1 -1 1 0 -3 10 -24 49 -89 145 -208 245 -174 -176 1121 -3185 7137 -13920 24301 -37926
Round trip:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37

Wren

Library: Wren-long

This hard-codes the sequences to be tested as I couldn't see much point in repeating code from the related tasks. 

import "./long" for Long, ULong

class BT {
    static forward(a) {
        var c = a.count
        var b = List.filled(c, null)
        for (n in 0...c) {
            var sum = Long.zero
            for (k in 0..n) sum = sum + ULong.binomial(n, k).toLong * a[k]
            b[n] = sum.toNum
        }
        return b
    }

    static inverse(b) {
        var c = b.count
        var a = List.filled(c, null)
        for (n in 0...c) {
            var sum = Long.zero
            for (k in 0..n) {
                var sign = (-1).pow(n-k)
                sum = sum + ULong.binomial(n, k).toLong * b[k] * sign
                a[n] = sum.toNum
            }
        }
        return a
    }
}

var seqs = [
    [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190],
    [0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0],
    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181],
    [1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37]
]

var names = [
    "Catalan number sequence:",
    "Prime flip-flop sequence:",
    "Fibonacci number sequence:",
    "Padovan number sequence:"
]

var fwd
for (i in 0...seqs.count) {
    System.print(names[i])
    System.print(seqs[i].join(" "))
    System.print("Forward binomial transform:")
    System.print((fwd = BT.forward(seqs[i])).join(" "))
    System.print("Inverse binomial transform:")
    System.print(BT.inverse(seqs[i]).join(" "))
    System.print("Round trip:")
    System.print(BT.inverse(fwd).join(" "))
    if (i < seqs.count - 1) System.print()
}
Output:
Catalan number sequence:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190
Forward binomial transform:
1 2 5 15 51 188 731 2950 12235 51822 223191 974427 4302645 19181100 86211885 390248055 1777495635 8140539950 37463689775 173164232965
Inverse binomial transform:
1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595
Round trip:
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670 129644790 477638700 1767263190

Prime flip-flop sequence:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0
Forward binomial transform:
0 1 3 6 11 20 37 70 134 255 476 869 1564 2821 5201 9948 19793 40562 84271 174952
Inverse binomial transform:
0 1 -1 0 3 -10 25 -56 118 -237 456 -847 1540 -2795 5173 -9918 19761 -40528 84235 -174914
Round trip:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0

Fibonacci number sequence:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
Forward binomial transform:
0 1 3 8 21 55 144 377 987 2584 6765 17711 46368 121393 317811 832040 2178309 5702887 14930352 39088169
Inverse binomial transform:
0 1 -1 2 -3 5 -8 13 -21 34 -55 89 -144 233 -377 610 -987 1597 -2584 4181
Round trip:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181

Padovan number sequence:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37
Forward binomial transform:
1 1 1 2 5 12 28 65 151 351 816 1897 4410 10252 23833 55405 128801 299426 696081 1618192
Inverse binomial transform:
1 -1 1 0 -3 10 -24 49 -89 145 -208 245 -174 -176 1121 -3185 7137 -13920 24301 -37926
Round trip:
1 0 0 1 0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37
Retrieved from "https://rosettacode.org/wiki/Binomial_transform?oldid=341661"