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A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(i*j*(j + 1)/2)).

%I A327744 #12 Sep 25 2019 01:25:55
%S A327744 1,1,2,4,6,9,17,23,35,54,77,108,163,221,309,436,593,800,1109,1470,
%T A327744 1968,2642,3482,4566,6052,7848,10204,13276,17092,21924,28245,35949,
%U A327744 45762,58231,73609,92789,117140,146799,183826,229995,286483,356040,442566,547489
%N A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(i*j*(j + 1)/2)).
%C A327744 Euler transform of A007862.
%H A327744 Vaclav Kotesovec, <a href="/A327744/b327744.txt">Table of n, a(n) for n = 0..10000</a>
%F A327744 G.f.: Product_{k>=1} 1 / (1 - x^k)^A007862(k).
%t A327744 nmax = 43; CoefficientList[Series[Product[1/(1 - x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
%t A327744 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]
%t A327744 nmax = 50; CoefficientList[Series[Product[1/QPochhammer[x^(k*(k + 1)/2)], {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 24 2019 *)
%Y A327744 Cf. A004101, A007862, A327745.
%K A327744 nonn
%O A327744 0,3
%A A327744 _Ilya Gutkovskiy_, Sep 23 2019