A182446 a(n) = Sum_{k = 0..n} C(n,k)^9.

1, 2, 514, 39368, 10601986, 2003906252, 588906874144, 159219918144128, 51207103076632066, 16425660314368351892, 5697191745563573732764, 2010823973962863400708688, 739753103704422167184400096, 277511604090132008416695054272, 106814999715696983804826836579584

Offset: 0

Author

Vaclav Kotesovec, Apr 29 2012

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vaclav Kotesovec, Recurrence (of order 5)
• 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..9)],[seq(1, i=1..8)],-1):
seq(simplify(a(n)),n=0..14); # Peter Luschny, Jul 27 2016

Mathematica

Table[Sum[Binomial[n, k]^9, {k, 0, n}], {n, 0, 25}]

PARI

a(n) = sum(k=0, n, binomial(n,k)^9); \\ Michel Marcus, Jul 17 2020

Formula

Asymptotic (p = 9): 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)^9*C(n,k)^9 = C(n,r)^9*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!)^9 = (Sum_{n>=0} x^n / (n!)^9)^2. - Ilya Gutkovskiy, Jul 17 2020