A182422 a(n) = Sum_{k = 0..n} C(n,k)^8.

1, 2, 258, 13124, 1810690, 200781252, 30729140484, 4579408029576, 770670360699138, 132018919625044100, 23913739057463037508, 4433505541977804821256, 848185646293853978499844, 165563367990287610967653512, 32993144260428865295508700680

Offset: 0

Author

Vaclav Kotesovec, Apr 28 2012

Keywords

• nonn
• easy

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.

Crossrefs

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..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 *)

PARI

a(n) = sum(k=0, n, binomial(n,k)^8); \\ Michel Marcus, Jul 17 2020

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