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%I #4 Jun 27 2018 13:15:16
%S 1,-1,0,-2,1,-2,3,-3,6,-8,14,-10,28,-26,41,-73,90,-112,155,-221,288,
%T -501,560,-799,1153,-1610,1953,-3095,4073,-5224,7295,-9536,13536,
%U -18402,24757,-32936,48714,-60790,82101,-113247,153330,-201522,275713,-367041,492991
%N Expansion of Product_{k>=1} 1/(1 + q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*q(j)^k*x^(j*k)/k).
%t nmax = 44; CoefficientList[Series[Product[1/(1 + PartitionsQ[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 44; CoefficientList[Series[Exp[Sum[Sum[(-1)^k PartitionsQ[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (-PartitionsQ[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 44}]
%Y Cf. A000009, A089254, A270995, A279785, A304786, A316230.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Jun 27 2018
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