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

1, 1, 2, 4, 9, 18, 37, 76, 158, 326, 672, 1386, 2862, 5906, 12187, 25148, 51900, 107103, 221023, 456110, 941256, 1942423, 4008481, 8272094, 17070712, 35227975, 72698206, 150023632, 309596255, 638898274, 1318462339, 2720844607, 5614870612, 11587126980

Offset: 0

Author

Ilya Gutkovskiy, Sep 23 2019

Keywords

• nonn

Comments

Invert transform of A046951.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A004101, A046951, A129921, A280451.

Programs

Maple

a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
      nops(select(issqr, numtheory[divisors](i))), i=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 23 2019

Mathematica

nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
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}]

Formula

G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^(k^2))).

G.f.: 1 / (1 - Sum_{k>=1} (theta_3(x^k) - 1) / 2), where theta_() is the Jacobi theta function.

a(0) = 1; a(n) = Sum_{k=1..n} A046951(k) * a(n-k).