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A182447
a(n) = Sum_{k = 0..n} C(n,k)^10.
12
1, 2, 1026, 118100, 62563330, 20019531252, 11393421713604, 5550455033938152, 3431955863873102850, 2052124795850957537060, 1367610300690018553312276, 916694195766256069610158152, 649630217578404016288230718276, 467800319852823195772146025385000
OFFSET
0,2
LINKS
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.
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
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}]
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)^10); \\ Michel Marcus, Jul 17 2020
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.
Sequence in context: A004802 A294279 A196292 * A218437 A166852 A236951
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Apr 29 2012
STATUS
approved