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A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).

%I A327738 #7 Sep 27 2019 15:16:52
%S A327738 1,1,2,4,9,18,37,76,158,326,672,1386,2862,5906,12187,25148,51900,
%T A327738 107103,221023,456110,941256,1942423,4008481,8272094,17070712,
%U A327738 35227975,72698206,150023632,309596255,638898274,1318462339,2720844607,5614870612,11587126980
%N A327738 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*j^2)).
%C A327738 Invert transform of A046951.
%H A327738 Alois P. Heinz, <a href="/A327738/b327738.txt">Table of n, a(n) for n = 0..1000</a>
%F A327738 G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^(k^2))).
%F A327738 G.f.: 1 / (1 - Sum_{k>=1} (theta_3(x^k) - 1) / 2), where theta_() is the Jacobi theta function.
%F A327738 a(0) = 1; a(n) = Sum_{k=1..n} A046951(k) * a(n-k).
%p A327738 a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
%p A327738       nops(select(issqr, numtheory[divisors](i))), i=1..n))
%p A327738     end:
%p A327738 seq(a(n), n=0..35);  # _Alois P. Heinz_, Sep 23 2019
%t A327738 nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
%t A327738 a[0] = 1; a[n_] := a[n] = Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]
%Y A327738 Cf. A004101, A046951, A129921, A280451.
%K A327738 nonn
%O A327738 0,3
%A A327738 _Ilya Gutkovskiy_, Sep 23 2019