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

%I #4 May 18 2018 19:48:31

%S 1,-1,-2,-1,-2,4,0,15,7,17,22,26,-79,-2,-12,-392,-250,-392,-443,-640,

%T -404,-795,5106,1147,3304,4542,32330,21001,23372,21015,14496,16165,

%U -17213,51296,-231330,-890169,-492310,-755449,-1648273,131600,-6308274,-2160440,-4410945,1593319

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

%C Convolution inverse of A063834.

%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 - A000041(k)*x^k).

%t nmax = 43; CoefficientList[Series[Product[(1 - PartitionsP[k] x^k), {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, 43}]

%Y Cf. A000041, A063834, A271619, A300508.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, May 18 2018