A336441 - OEIS

%I #7 Oct 28 2024 04:39:15

%S 1,1,3,71,30232,435772624,357189846148256,25740403176657987904960,

%T 234446578865185870182814945640448,

%U 363178754511398964104990417951192651478859776,122088173887703514886799765831338556792096849201928981184512

%N a(n) = (n!)^n * [x^n] exp(Sum_{k>=1} x^k / k^n).

%F From _Vaclav Kotesovec_, Oct 28 2024: (Start)

%F a(n) ~ (n!)^(n-1).

%F a(n) ~ (2*Pi)^((n-1)/2) * n^(n^2 - n/2 - 1/2) / exp(n^2 - n - 1/12). (End)

%t Table[(n!)^n SeriesCoefficient[Exp[Sum[x^k/k^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]

%t b[n_, k_] := If[n == 0, 1, (1/n) Sum[(Binomial[n, j] (n - j - 1)!)^k (n - j) b[j, k], {j, 0, n - 1}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]

%Y Cf. A000142, A074707, A193436, A217145, A275044.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 21 2020