A295612 a(n) = Sum_{k=0..n} binomial(n+k,k)^k.

1, 3, 40, 8105, 24053106, 1016507243472, 622366942086680904, 5608321882919220905812521, 752711651805019773658037206391596, 1518219710649896586598445898967340890577318, 46343146356260529633020448755386347142785083052620084

Offset: 0

Author

Ilya Gutkovskiy, Nov 24 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..41
• Eric Weisstein's World of Mathematics, Binomial Sums

Crossrefs

Cf. A001700, A046899, A112028, A112029, A167008, A219562, A219563, A219564.

Programs

Mathematica

Table[Sum[Binomial[n + k, k]^k, {k, 0, n}], {n, 0, 10}]
Table[Sum[((n + k)!/(n! k!))^k, {k, 0, n}], {n, 0, 10}]

PARI

a(n) = sum(k=0, n, binomial(n+k,k)^k); \\ Michel Marcus, Nov 25 2017

Formula

a(n) = Sum_{k=0..n} A046899(n,k)^k.

a(n) ~ 2^(2*n^2) / (exp(1/8) * Pi^(n/2) * n^(n/2)). - Vaclav Kotesovec, Nov 25 2017