A321520 Expansion of Product_{k>=1} (1 + (k - 1)!*x^k).

1, 1, 1, 3, 8, 32, 152, 882, 5964, 46644, 411564, 4056912, 44097072, 524234448, 6761911968, 94055452128, 1403047948320, 22342552398720, 378256278306240, 6783950610708480, 128480976137122560, 2562250754919421440, 53668564630447910400

Offset: 0

Author

Ilya Gutkovskiy, Nov 12 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A107107, A265950.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 05 2023

Mathematica

nmax = 22; CoefficientList[Series[Product[(1 + (k - 1)! x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d ((d - 1)!)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 22}]

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*((d - 1)!)^(k/d) ) * x^k/k).

a(n) ~ (n-1)! * (1 + 1/n + 2/n^2 + 7/n^3 + 34/n^4 + 203/n^5 + 1454/n^6 + 12321/n^7 + 121326/n^8 + 1364947/n^9 + 17301550/n^10 + ...). - Vaclav Kotesovec, Nov 13 2018