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%I #4 Mar 21 2018 19:25:19
%S 1,75,272880,4681655040,221478589107480,22313622005672849712,
%T 4108665216956980742226192,1249503956658157724969373808320,
%U 583952821303314451291898006535866460,397372225886096887788939487944785734626120,377577476850495509525002042506806447493291890064
%N a(n) = Sum_{k>=0} binomial(k^4, n)/2^(k+1).
%C 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)).
%F a(n) ~ 2^(8*n) * n^(3*n) / (exp(3*n) * (log(2))^(4*n+1)).
%t Table[Sum[Binomial[k^4, n]/2^(k+1), {k, 0, Infinity}], {n, 0, 12}]
%t Table[Sum[StirlingS1[n, j] * HurwitzLerchPhi[1/2, -4*j, 0]/2, {j, 0, n}] / n!, {n, 0, 12}]
%Y Cf. A173217 (m=2), A301466 (m=3), A301310.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Mar 21 2018