A305152 Expansion of Sum_{k>0} x^(k^2) / (1 + x^k).

1, -1, 1, 0, 1, -2, 1, 0, 2, -2, 1, -1, 1, -2, 2, 1, 1, -3, 1, -1, 2, -2, 1, 0, 2, -2, 2, -1, 1, -4, 1, 1, 2, -2, 2, -1, 1, -2, 2, 0, 1, -4, 1, -1, 3, -2, 1, 1, 2, -3, 2, -1, 1, -4, 2, 0, 2, -2, 1, -2, 1, -2, 3, 2, 2, -4, 1, -1, 2, -4, 1, 0, 1, -2, 3, -1, 2, -4, 1

Offset: 1

Author

Seiichi Manyama, May 26 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A038548, A048272, A193773 (odd bisection), A348608, A228441, A010052, A348952.

Programs

PARI

{a(n) = polcoeff(sum(k=1, sqrtint(n), x^(k^2)/(1+x^k))+x*O(x^n), n)}

PARI

a(n) = sumdiv(n, d, if (d <= sqrtint(n), (-1)^(d + n/d))); \\ Michel Marcus, Nov 03 2021

Formula

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d). - Ilya Gutkovskiy, Nov 02 2021

a(n) = (A228441(n) + A010052(n))/2. - Ridouane Oudra, Aug 14 2025

a(n) = A010052(n) - A348952(n). - Ridouane Oudra, Aug 20 2025