cp's OEIS Frontend

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

%I A321520 #14 Dec 28 2023 07:09:29
%S A321520 1,1,1,3,8,32,152,882,5964,46644,411564,4056912,44097072,524234448,
%T A321520 6761911968,94055452128,1403047948320,22342552398720,378256278306240,
%U A321520 6783950610708480,128480976137122560,2562250754919421440,53668564630447910400
%N A321520 Expansion of Product_{k>=1} (1 + (k - 1)!*x^k).
%H A321520 Alois P. Heinz, <a href="/A321520/b321520.txt">Table of n, a(n) for n = 0..450</a>
%F A321520 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*((d - 1)!)^(k/d) ) * x^k/k).
%F A321520 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
%p A321520 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A321520      `if`(n=0, 1, b(n, i-1)+b(n-i, min(n-i, i-1))*(i-1)!))
%p A321520     end:
%p A321520 a:= n-> b(n$2):
%p A321520 seq(a(n), n=0..24);  # _Alois P. Heinz_, Jul 05 2023
%t A321520 nmax = 22; CoefficientList[Series[Product[(1 + (k - 1)! x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A321520 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}]
%Y A321520 Cf. A107107, A265950.
%K A321520 nonn
%O A321520 0,4
%A A321520 _Ilya Gutkovskiy_, Nov 12 2018