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%I #4 May 18 2018 19:48:23
%S 1,-1,-1,-2,0,-1,2,3,11,8,19,13,22,-5,-10,-80,-105,-246,-303,-502,
%T -506,-681,-400,-231,873,1956,4733,7536,12891,17609,25188,29508,34890,
%U 29690,19039,-17742,-74002,-183563,-333665,-572271,-866683,-1271429,-1698491,-2181207
%N Expansion of Product_{k>=1} 1/(1 + x^k)^p(k), where p(k) = number of partitions of k (A000041).
%C Convolution inverse of A261049.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: Product_{k>=1} 1/(1 + x^k)^A000041(k).
%t nmax = 43; CoefficientList[Series[Product[1/(1 + x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d PartitionsP[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]
%Y Cf. A000041, A001970, A089254, A261049, A300508.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, May 18 2018