A005261 a(n) = Sum_{k = 0..n} C(n,k)^5.

1, 2, 34, 488, 9826, 206252, 4734304, 113245568, 2816649826, 72001228052, 1883210876284, 50168588906768, 1357245464138656, 37198352117916992, 1030920212982957184, 28847760730478655488, 814066783370083977826

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the constant term in the expansion of ((1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n. - Seiichi Manyama, Oct 27 2019

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
• M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.
• Eric Weisstein's World of Mathematics, Binomial sums.
• Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes.
• Jin Yuan, Zhi-Juan Lu, Asmus L. Schmidt, On recurrences for sums of powers of binomial coefficients, J. Numb. Theory 128 (2008) 2784-2794

Crossrefs

Column k=5 of A309010.

Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.

Programs

Maple

a := n -> hypergeom([seq(-n, i=1..5)],[seq(1, i=1..4)], -1):
seq(simplify(a(n)),n=0..16); # Peter Luschny, Jul 27 2016

Mathematica

RecurrenceTable[{32*(55n^2+33n+6)*(n-1)^4*a[n-2]-(19415n^6-27181n^5+7453n^4+3289n^3-956n^2-276n+96)*a[n-1]-(1155n^6+693n^5-732n^4-715n^3+45n^2+210n+56)*a[n]+(55n^2-77n+28)*(n+1)^4*a[n+1]==0,a[0]==1,a[1]==2,a[2]==34},a,{n,0,25}]
(* or directly *)
Table[Sum[Binomial[n,k]^5,{k,0,n}],{n,0,25}] (* Vaclav Kotesovec, Apr 27 2012 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)^5); \\ Michel Marcus, Mar 09 2016

Python

def A005261(n):
    m, g = 1, 0
    for k in range(n+1):
        g += m
        m = m*(n-k)**5//(k+1)**5
    return g # Chai Wah Wu, Oct 04 2022

Formula

a(n) ~ 4*5^(-1/2)*Pi^-2*n^-2*2^(5*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002

Recurrence (M. A. Perlstadt, 1987): 32*(55*n^2 + 33*n + 6)*(n - 1)^4*a(n-2) - (19415*n^6 - 27181*n^5 + 7453*n^4 + 3289*n^3 - 956*n^2 - 276*n + 96)*a(n-1) - (1155*n^6 + 693*n^5 - 732*n^4 - 715*n^3 + 45*n^2 + 210*n + 56)*a(n) + (55*n^2 - 77*n + 28)*(n + 1)^4*a(n+1) = 0. [Vaclav Kotesovec, Apr 27 2012]

For r a nonnegative integer, Sum_{k = r..n} C(k,r)^5*C(n,k)^5 = C(n,r)^5*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016

Sum_{n>=0} a(n) * x^n / (n!)^5 = (Sum_{n>=0} x^n / (n!)^5)^2. - Ilya Gutkovskiy, Jul 17 2020

From Peter Bala, Nov 01 2024: (Start)

For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^4 * binomial(n-1, k).

For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n, -n, -n + 1], [1, 1, 1, 1], -1). (End)

Extensions

More terms from Matthew Conroy, Mar 16 2006