A276516 Expansion of Product_{k>=1} (1-x^(k^2)).

1, -1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 1, -1, 0, -1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, -1, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 1, 1, 1, 0, 0, -1, -1, 1, 1, -1, 0, 0, -1, 1, -1, 2, -1, 0, 1, -2, 0, 1, 0, 1, 0, -1, 0, -2, 2, 0

Offset: 0

Author

Vaclav Kotesovec, Dec 12 2016

Keywords

• sign
• look

Comments

The difference between the number of partitions of n into an even number of distinct squares and the number of partitions of n into an odd number of distinct squares. - Ilya Gutkovskiy, Jan 15 2018

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.

Crossrefs

Cf. A001156, A033461, A279360, A279368, A276517, A341040.

Programs

Mathematica

nn = 15; CoefficientList[Series[Product[(1-x^(k^2)), {k, nn}], {x, 0, nn^2}], x]
nmax = 200; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; Take[poly, nmax+1]

Formula

a(n) = Sum_{k>=0} (-1)^k * A341040(n,k). - Alois P. Heinz, Feb 03 2021

a(n) = A033461(n) - 2*A339367(n). - R. J. Mathar, Jul 29 2025