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A321522 Expansion of Product_{k>=1} (1 + x^k)^((k-1)!).

%I A321522 #13 Dec 28 2023 07:08:51
%S A321522 1,1,1,3,8,32,153,883,5980,46660,411861,4057263,44104688,524243696,
%T A321522 6762188285,94055795999,1403061499362,22342571084082,378257158227079,
%U A321522 6783952072695685,128481050502464062,2562250926987454694,53668572808754641369,1177957644341460946099
%N A321522 Expansion of Product_{k>=1} (1 + x^k)^((k-1)!).
%H A321522 Alois P. Heinz, <a href="/A321522/b321522.txt">Table of n, a(n) for n = 0..450</a>
%F A321522 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d! ) * x^k/k).
%F A321522 a(n) ~ (n-1)! * (1 + 1/n + 2/n^2 + 7/n^3 + 34/n^4 + 203/n^5 + 1455/n^6 + 12343/n^7 + 121636/n^8 + 1368647/n^9 + 17343274/n^10 + ...). - _Vaclav Kotesovec_, Nov 13 2018
%p A321522 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A321522       add(b(n-i*j, i-1)*binomial((i-1)!, j), j=0..n/i)))
%p A321522     end:
%p A321522 a:= n-> b(n$2):
%p A321522 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 10 2021
%t A321522 nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x]
%t A321522 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 23}]
%Y A321522 Cf. A000142, A051295, A104150, A179327, A261052, A321521.
%K A321522 nonn
%O A321522 0,4
%A A321522 _Ilya Gutkovskiy_, Nov 12 2018