A005261 a(n) = Sum_{k = 0..n} C(n,k)^5. (Formerly M2156)

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

Offset

0,2

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

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 1 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)

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 *)

Prog

(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

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.

Sequence in context: A092408 A180764 A228654 * A104898 A218432 A071799

Adjacent sequences: A005258 A005259 A005260 * A005262 A005263 A005264

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Matthew Conroy, Mar 16 2006

Status

approved