A336214 a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.

1, 1, 8, 270, 41984, 30706250, 94770093312, 1336016204844832, 76829717664330940416, 19838680914222199482800274, 20521247958509575370600000000000, 94285013320530947020636486516362047300, 1715947732437668013396578734960052732361179136

Offset: 0

Author

Vaclav Kotesovec, Jul 12 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..56

Crossrefs

Cf. A072034, A086331, A167008, A167010, A328812, A336188, A336204, A336212, A336213.

Programs

Mathematica

Flatten[{1, Table[Sum[k^n*Binomial[n, k]^n, {k, 1, n}], {n, 1, 15}]}]

PARI

a(n) = if (n==0, 1, sum(k=0, n, k^n * binomial(n,k)^n)); \\ Michel Marcus, Jul 13 2020

Formula

a(n) ~ c * exp(-1/4) * 2^(n^2 - n/2) * n^(n/2) / Pi^(n/2), where c = Sum_{k = -infinity..infinity} exp(-2*k*(k-1)) = exp(1/2) * sqrt(Pi/2) * EllipticTheta(3, -Pi/2, exp(-Pi^2/2)) = 2.036643566277677716389243890291939003151565... if n is even and c = Sum_{k = -infinity..infinity} exp(-2*k^2 + 1/2) = exp(1/2) * EllipticTheta(3, 0, exp(-2)) = 2.096087809957308346119920713317351288828811... if n is odd.

a(n) = n^n * A328812(n-1) for n > 0. - Seiichi Manyama, Jul 15 2020