A340103 - OEIS

%I #22 May 09 2021 05:37:34

%S 1,1,2,12,80,875,10584,170471,2949120,63772920,1441000000,38818444632,

%T 1089573617664,35185728919614,1175820172477440,44425722744140625,

%U 1722925924631969792,74364737115532234518,3291298649632850485248,159785357022861166517580,7932051456000000000000000

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

%H Vaclav Kotesovec, <a href="/A340103/b340103.txt">Table of n, a(n) for n = 0..385</a>

%F 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.

%F a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - _Vaclav Kotesovec_, May 09 2021

%t Table[SeriesCoefficient[Product[(1 + n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

%t 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}]

%t Join[{1}, Table[SeriesCoefficient[n*QPochhammer[-1/n, n*x]/(n+1), {x, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, May 09 2021 *)

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 24 2021