A301468 a(n) = Sum_{k>=0} binomial(k^4, n)/2^(k+1).

1, 75, 272880, 4681655040, 221478589107480, 22313622005672849712, 4108665216956980742226192, 1249503956658157724969373808320, 583952821303314451291898006535866460, 397372225886096887788939487944785734626120, 377577476850495509525002042506806447493291890064

Offset: 0

Author

Vaclav Kotesovec, Mar 21 2018

Keywords

• nonn

Comments

In general, for m > 2, Sum_{k>=0} binomial(k^m, n) / 2^(k+1) is asymptotic to m^(m*n + 1/2) * n^((m-1)*n) / (2*exp((m-1)*n) * (log(2))^(m*n + 1)).

Crossrefs

Cf. A173217 (m=2), A301466 (m=3), A301310.

Programs

Mathematica

Table[Sum[Binomial[k^4, n]/2^(k+1), {k, 0, Infinity}], {n, 0, 12}]
Table[Sum[StirlingS1[n, j] * HurwitzLerchPhi[1/2, -4*j, 0]/2, {j, 0, n}] / n!, {n, 0, 12}]

Formula

a(n) ~ 2^(8*n) * n^(3*n) / (exp(3*n) * (log(2))^(4*n+1)).