A182447 a(n) = Sum_{k = 0..n} C(n,k)^10.

1, 2, 1026, 118100, 62563330, 20019531252, 11393421713604, 5550455033938152, 3431955863873102850, 2052124795850957537060, 1367610300690018553312276, 916694195766256069610158152, 649630217578404016288230718276, 467800319852823195772146025385000

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..10)],[seq(1, i=1..9)],1):
seq(simplify(a(n)),n=0..13); # Peter Luschny, Jul 27 2016

Mathematica

Table[Sum[Binomial[n, k]^10, {k, 0, n}], {n, 0, 25}]

PARI

a(n) = sum(k=0, n, binomial(n,k)^10); \\ Michel Marcus, Jul 17 2020

Formula

Asymptotic (p = 10): 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)^10*C(n,k)^10 = C(n,r)^10*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!)^10 = (Sum_{n>=0} x^n / (n!)^10)^2. - Ilya Gutkovskiy, Jul 17 2020