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A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j*(j + 1)/2)).

%I A327745 #9 Sep 24 2019 15:28:33
%S A327745 1,1,1,3,3,4,8,9,11,19,23,28,42,51,62,89,108,130,178,215,260,344,413,
%T A327745 496,639,766,916,1155,1380,1641,2040,2426,2870,3520,4166,4912,5960,
%U A327745 7023,8246,9911,11634,13610,16224,18972,22111,26183,30507,35430,41698
%N A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j*(j + 1)/2)).
%C A327745 Weigh transform of A007862.
%H A327745 Vaclav Kotesovec, <a href="/A327745/b327745.txt">Table of n, a(n) for n = 0..10000</a>
%F A327745 G.f.: Product_{k>=1} (1 + x^k)^A007862(k).
%t A327745 nmax = 48; CoefficientList[Series[Product[(1 + x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
%t A327745 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]
%t A327745 nmax = 50; CoefficientList[Series[Product[QPochhammer[-1, x^(k*(k + 1)/2)]/2, {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 24 2019 *)
%Y A327745 Cf. A007862, A280451, A327744.
%K A327745 nonn
%O A327745 0,4
%A A327745 _Ilya Gutkovskiy_, Sep 23 2019