A304783 - OEIS

%I #4 May 18 2018 19:48:16

%S 1,-1,-1,-1,0,1,0,3,2,3,1,3,-2,0,-6,-8,-12,-14,-18,-19,-19,-15,-3,4,

%T 29,46,90,114,165,192,248,252,276,232,185,29,-143,-454,-811,-1324,

%U -1909,-2609,-3348,-4132,-4851,-5386,-5653,-5380,-4470,-2477,664,5582,12193,21314

%N Expansion of Product_{k>=1} (1 - x^k)^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

%C Convolution inverse of A089259.

%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.: Product_{k>=1} (1 - x^k)^A000009(k).

%t nmax = 53; CoefficientList[Series[Product[(1 - x^k)^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 53}]

%Y Cf. A000009, A050342, A089254, A089259, A300508.

%K sign

%O 0,8

%A _Ilya Gutkovskiy_, May 18 2018