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%I #6 Aug 10 2018 17:32:01
%S 1,0,2,3,8,13,31,53,112,201,393,710,1343,2409,4431,7912,14255,25223,
%T 44787,78519,137700,239347,415343,716001,1231326,2106287,3593141,
%U 6102679,10335269,17437476,29337139,49192762,82261930,137148782,228061165,378198633,625623318,1032301633
%N Expansion of Product_{k>=2} 1/(1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).
%C First differences of A001970.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: exp(Sum_{j>=1} Sum_{k>=2} p(k)*x^(j*k)/j).
%p with(numtheory): with(combinat):
%p b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
%p end:
%p a:= n-> b(n)-b(n-1):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 10 2018
%t nmax = 37; CoefficientList[Series[Product[1/(1 - x^k)^PartitionsP[k], {k, 2, nmax}], {x, 0, nmax}], x]
%t nmax = 37; CoefficientList[Series[Exp[Sum[Sum[PartitionsP[k] x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
%t b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d PartitionsP[d], {d, Divisors[k]}] b[n - k], {k, 1, n}]/n]; Differences[Table[b[n], {n, -1, 37}]]
%Y Cf. A000041, A001970, A002865, A191659, A291552.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 10 2018
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