A316230 - OEIS

%I #5 Jun 27 2018 13:15:09

%S 1,-1,-1,-2,1,-2,2,0,24,-17,31,-21,94,-107,121,-443,742,-977,532,

%T -2159,3275,-6193,6988,-11156,30278,-39214,42759,-80255,149070,

%U -193093,291229,-451125,1017812,-1335848,1609412,-3248202,5606551,-7684574,10012531,-17908468

%N Expansion of Product_{k>=1} 1/(1 + p(k)*x^k), where p(k) = number of partitions of k (A000041).

%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.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*p(j)^k*x^(j*k)/k).

%t nmax = 39; CoefficientList[Series[Product[1/(1 + PartitionsP[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 39; CoefficientList[Series[Exp[Sum[Sum[(-1)^k PartitionsP[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 (-PartitionsP[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 39}]

%Y Cf. A000041, A063834, A271619, A304784, A304785, A316231.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Jun 27 2018