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A317911 Expansion of Product_{k>=2} 1/(1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).

%I A317911 #6 Aug 10 2018 17:32:01
%S A317911 1,0,2,3,8,13,31,53,112,201,393,710,1343,2409,4431,7912,14255,25223,
%T A317911 44787,78519,137700,239347,415343,716001,1231326,2106287,3593141,
%U A317911 6102679,10335269,17437476,29337139,49192762,82261930,137148782,228061165,378198633,625623318,1032301633
%N A317911 Expansion of Product_{k>=2} 1/(1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).
%C A317911 First differences of A001970.
%H A317911 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A317911 G.f.: exp(Sum_{j>=1} Sum_{k>=2} p(k)*x^(j*k)/j).
%p A317911 with(numtheory): with(combinat):
%p A317911 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A317911       numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A317911     end:
%p A317911 a:= n-> b(n)-b(n-1):
%p A317911 seq(a(n), n=0..40);  # _Alois P. Heinz_, Aug 10 2018
%t A317911 nmax = 37; CoefficientList[Series[Product[1/(1 - x^k)^PartitionsP[k], {k, 2, nmax}], {x, 0, nmax}], x]
%t A317911 nmax = 37; CoefficientList[Series[Exp[Sum[Sum[PartitionsP[k] x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
%t A317911 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 A317911 Cf. A000041, A001970, A002865, A191659, A291552.
%K A317911 nonn
%O A317911 0,3
%A A317911 _Ilya Gutkovskiy_, Aug 10 2018