OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Recurrence (of order 4)
M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.
FORMULA
Asymptotic (p = 8): a(n) ~ 2^(p*n)/sqrt(p)*(2/(Pi*n))^((p - 1)/2)*( 1 - (p - 1)^2/(4*p*n) + O(1/n^2) ).
For r a nonnegative integer, Sum_{k = r..n} C(k,r)^8*C(n,k)^8 = C(n,r)^8*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!)^8 = (Sum_{n>=0} x^n / (n!)^8)^2. - Ilya Gutkovskiy, Jul 17 2020
MAPLE
a := n -> hypergeom([seq(-n, i=1..8)], [seq(1, i=1..7)], 1):
seq(simplify(a(n)), n=0..14); # Peter Luschny, Jul 27 2016
MATHEMATICA
Table[Total[Binomial[n, Range[0, n]]^8], {n, 0, 20}] (* T. D. Noe, Apr 28 2012 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)^8); \\ Michel Marcus, Jul 17 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Apr 28 2012
STATUS
approved