A292520 Expansion of Product_{k>=1} 1/(1 + x^(k^2)).

1, -1, 1, -1, 0, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, 0, -1, 2, -2, 2, -1, 0, 0, 0, -1, 2, -3, 3, -2, 1, 0, 1, -2, 3, -4, 3, -2, 1, 0, 1, -2, 3, -4, 3, -2, 1, 0, 0, -2, 4, -5, 6, -4, 2, -1, 0, -2, 5, -7, 8, -6, 3, -1, 0, -1, 3, -6, 7, -6, 4, -1, 1, -1, 3, -6, 7, -8, 6, -3, 2, -4, 6, -9, 11, -9, 7, -4, 1, -3, 7

Offset: 0

Author

Ilya Gutkovskiy, Sep 18 2017

Keywords

• sign

Comments

Convolution inverse of A033461.

The difference between the number of partitions of n into an even number of squares and the number of partitions of n into an odd number of squares.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..20000
• Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
• Index entries for related partition-counting sequences

Crossrefs

Cf. A001156, A033461, A081362, A243148, A276516, A279225, A279226.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1 + x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 19 2017 *)

Formula

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

a(n) ~ (-1)^n * exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(7/3)) * Zeta(3/2)^(1/3) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 19 2017

a(n) = Sum_{k=0..n} (-1)^k * A243148(n,k). - Alois P. Heinz, Jul 25 2022