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A304966 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-1), where p(k) = number of partitions of k (A000041).

%I A304966 #6 May 22 2018 20:34:03
%S A304966 1,0,1,2,5,8,18,30,61,107,203,358,663,1162,2093,3666,6481,11258,19652,
%T A304966 33874,58464,100046,171032,290563,492745,831393,1399655,2346707,
%U A304966 3924873,6541472,10875575,18025629,29804125,49143254,80841455,132651457,217179366,354745107,578215807
%N A304966 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-1), where p(k) = number of partitions of k (A000041).
%C A304966 Euler transform of A000065.
%C A304966 Convolution of the sequences A001970 and A010815.
%H A304966 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A304966 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A304966 G.f.: Product_{k>=1} 1/(1 - x^k)^A000065(k).
%p A304966 with(combinat): with(numtheory):
%p A304966 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A304966       numbpart(d)-d, d=divisors(j))*a(n-j), j=1..n)/n)
%p A304966     end:
%p A304966 seq(a(n), n=0..40);  # _Alois P. Heinz_, May 22 2018
%t A304966 nmax = 38; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
%t A304966 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]
%Y A304966 Cf. A000041, A000065, A000219, A001383, A001970, A010815, A052847.
%K A304966 nonn
%O A304966 0,4
%A A304966 _Ilya Gutkovskiy_, May 22 2018