A340103 a(n) = [x^n] Product_{k>=1} (1 + n^(k-1)*x^k).

1, 1, 2, 12, 80, 875, 10584, 170471, 2949120, 63772920, 1441000000, 38818444632, 1089573617664, 35185728919614, 1175820172477440, 44425722744140625, 1722925924631969792, 74364737115532234518, 3291298649632850485248, 159785357022861166517580, 7932051456000000000000000

Offset: 0

Author

Ilya Gutkovskiy, Apr 24 2021

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..385

Crossrefs

Cf. A000009, A003056, A008289, A291698, A292305, A304961, A338697, A344094.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 + n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] n^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 20}]
Join[{1}, Table[SeriesCoefficient[n*QPochhammer[-1/n, n*x]/(n+1), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)

Formula

a(n) = Sum_{k=0..A003056(n)} q(n,k) * n^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - Vaclav Kotesovec, May 09 2021