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

%I A276516 #38 Jul 29 2025 12:27:05
%S A276516 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,
%T A276516 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,
%U A276516 0,-1,1,-1,2,-1,0,1,-2,0,1,0,1,0,-1,0,-2,2,0
%N A276516 Expansion of Product_{k>=1} (1-x^(k^2)).
%C A276516 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
%H A276516 Vaclav Kotesovec, <a href="/A276516/b276516.txt">Table of n, a(n) for n = 0..10000</a>
%H A276516 Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.
%F A276516 a(n) = Sum_{k>=0} (-1)^k * A341040(n,k). - _Alois P. Heinz_, Feb 03 2021
%F A276516 a(n) = A033461(n) - 2*A339367(n). - _R. J. Mathar_, Jul 29 2025
%t A276516 nn = 15; CoefficientList[Series[Product[(1-x^(k^2)), {k, nn}], {x, 0, nn^2}], x]
%t A276516 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]
%Y A276516 Cf. A001156, A033461, A279360, A279368, A276517, A341040.
%K A276516 sign,look
%O A276516 0,50
%A A276516 _Vaclav Kotesovec_, Dec 12 2016