A329971 Expansion of 1 / (1 - 2 * Sum_{k>=1} x^(k^2)).

1, 2, 4, 8, 18, 40, 88, 192, 420, 922, 2024, 4440, 9736, 21352, 46832, 102720, 225298, 494144, 1083804, 2377112, 5213736, 11435312, 25081112, 55010496, 120654744, 264632554, 580419672, 1273036832, 2792156864, 6124049048, 13431901808, 29460245120, 64615275940

Offset: 0

Author

Ilya Gutkovskiy, Nov 26 2019

Keywords

• nonn

Crossrefs

Cf. A000122, A004402, A006456, A025192, A032803, A240944, A279225, A279226, A280542, A320654.

Programs

Mathematica

nmax = 32; CoefficientList[Series[1/(1 - 2 Sum[x^(k^2), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

Formula

G.f.: 1 / (2 - theta_3(x)), where theta_3() is the Jacobi theta function.

a(0) = 1; a(n) = Sum_{k=1..n} A000122(k) * a(n-k).