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A304967 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-p(k-1)), where p(k) = number of partitions of k (A000041).

%I A304967 #7 May 22 2018 20:34:10
%S A304967 1,0,1,1,3,3,8,9,20,26,49,68,123,173,295,432,707,1044,1672,2483,3900,
%T A304967 5817,8993,13424,20539,30609,46399,69052,103879,154198,230550,341261,
%U A304967 507484,749028,1108559,1631340,2404311,3527615,5179317,7577263,11086413,16173577,23588227
%N A304967 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-p(k-1)), where p(k) = number of partitions of k (A000041).
%C A304967 Euler transform of A002865.
%H A304967 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A304967 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A304967 G.f.: Product_{k>=1} 1/(1 - x^k)^A002865(k).
%p A304967 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A304967       (numtheory[sigma](j)-1)*b(n-j), j=1..n)/n)
%p A304967     end:
%p A304967 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A304967       b(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A304967     end:
%p A304967 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 22 2018
%t A304967 nmax = 42; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - PartitionsP[k - 1]), {k, 1, nmax}], {x, 0, nmax}], x]
%t A304967 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - PartitionsP[d - 1]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 42}]
%Y A304967 Cf. A000041, A000219, A001383, A001970, A002865, A304966.
%K A304967 nonn
%O A304967 0,5
%A A304967 _Ilya Gutkovskiy_, May 22 2018