A320287 a(n) = n! * [x^n] Sum_{k>=0} exp(n^k*x)*x^k/k!.

1, 2, 6, 56, 2050, 318752, 252035714, 980755711616, 23647746367946754, 3088949241542073508352, 2940240000900000020000000002, 16218429504693724464229916894517248, 748528620411995327278028288988088683724802, 210422023062476527874650307058798916093350502080512

Offset: 0

Author

Ilya Gutkovskiy, Oct 09 2018

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..48

Crossrefs

Cf. A047863, A135079.

Programs

Magma

[(&+[Binomial(n,k)*n^(k*(n-k)):k in [0..n]]): n in [0..20]]; // G. C. Greubel, Nov 04 2018

Mathematica

Join[{1}, Table[n! SeriesCoefficient[Sum[Exp[n^k x] x^k/k!, {k, 0, n}], {x, 0, n}], {n, 13}]]
Join[{1}, Table[SeriesCoefficient[Sum[x^k/(1 - n^k x)^(k + 1), {k, 0, n}], {x, 0, n}], {n, 13}]]
Join[{1}, Table[Sum[Binomial[n, k] n^(k (n - k)), {k, 0, n}], {n, 13}]]

PARI

for(n=0,20, print1(sum(k=0,n, binomial(n,k)*n^(k*(n-k))), ", ")) \\ G. C. Greubel, Nov 04 2018

Formula

a(n) = [x^n] Sum_{k>=0} x^k/(1 - n^k*x)^(k+1).

a(n) = Sum_{k=0..n} binomial(n,k)*n^(k*(n-k)).

a(n) ~ 2^(n + 1/2) * n^(n^2/4 - 1/2) / sqrt(Pi) if n is even and a(n) ~ 2^(n + 3/2) * n^(n^2/4 - 3/4) / sqrt(Pi) if n is odd. - Vaclav Kotesovec, Jul 06 2022