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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Matthew Conroy, Mar 16 2006
STATUS
approved