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A305651 Expansion of Product_{k>=1} (1 + x^k)^(q(k)-1), where q(k) = number of partitions of k into distinct parts (A000009).

%I A305651 #9 Jun 09 2018 00:31:18
%S A305651 1,0,0,1,1,2,3,5,7,12,17,26,39,59,87,132,192,284,419,612,892,1303,
%T A305651 1887,2730,3945,5677,8154,11689,16711,23839,33960,48244,68432,96888,
%U A305651 136922,193148,272058,382508,537007,752735,1053550,1472406,2054988,2863993,3986245,5541008
%N A305651 Expansion of Product_{k>=1} (1 + x^k)^(q(k)-1), where q(k) = number of partitions of k into distinct parts (A000009).
%C A305651 Weigh transform of A111133.
%C A305651 Convolution of the sequences A050342 and A081362.
%H A305651 Alois P. Heinz, <a href="/A305651/b305651.txt">Table of n, a(n) for n = 0..1000</a>
%H A305651 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305651 G.f.: Product_{k>=1} (1 + x^k)^A111133(k).
%F A305651 G.f.: Product_{k>=1} (1 + x^k)^(A000009(k)-1).
%p A305651 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A305651       `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A305651     end:
%p A305651 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A305651       add(binomial(g(i)-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p A305651     end:
%p A305651 a:= n-> b(n$2):
%p A305651 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jun 07 2018
%t A305651 nmax = 45; CoefficientList[Series[Product[(1 + x^k)^(PartitionsQ[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
%t A305651 nmax = 45; CoefficientList[Series[Exp[Sum[(-1)^(k + 1)/k (1/ QPochhammer[x^k, x^(2 k)] - 1/(1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A305651 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (PartitionsQ[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]
%Y A305651 Cf. A000009, A050342, A081362, A089259, A111133, A304966, A304969.
%K A305651 nonn
%O A305651 0,6
%A A305651 _Ilya Gutkovskiy_, Jun 07 2018