A279368 -id:A279368 - cp's OEIS frontend

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

Vaclav Kotesovec, Dec 12 2016

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

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.

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

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]

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

A279360 Expansion of Product_{k>=1} (1+2*x^(k^2)).

Original entry on oeis.org

1, 2, 0, 0, 2, 4, 0, 0, 0, 2, 4, 0, 0, 4, 8, 0, 2, 4, 0, 0, 4, 8, 0, 0, 0, 6, 12, 0, 0, 12, 24, 0, 0, 0, 4, 8, 2, 4, 8, 16, 4, 12, 8, 0, 0, 12, 24, 0, 0, 10, 28, 16, 4, 12, 24, 32, 8, 16, 4, 8, 0, 12, 32, 16, 2, 32, 56, 0, 4, 16, 24, 16, 0, 4, 36, 56, 0, 16

Offset: 0

Vaclav Kotesovec, Dec 10 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001422, A032302, A033461, A279226, A279368, A341040.

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

a(n) ~ c^(1/3) * exp(3 * 2^(-4/3) * c^(2/3) * Pi^(1/3) * n^(1/3)) / (3 * 2^(2/3) * Pi^(1/3) * n^(5/6)), where c = -PolyLog(3/2, -2) = 1.28138038315976963883198... . - Vaclav Kotesovec, Dec 12 2016
From Alois P. Heinz, Feb 03 2021: (Start)
a(n) = Sum_{k>=0} 2^k * A341040(n,k).
a(n) = 0 <=> n in { A001422 }. (End)

A280225 G.f.: Product_{k>=1} (1 + 3*x^(k^2)) / (1-x^k).

Original entry on oeis.org

1, 4, 5, 9, 17, 34, 47, 75, 109, 165, 240, 341, 473, 671, 936, 1268, 1722, 2325, 3091, 4099, 5403, 7083, 9207, 11923, 15339, 19682, 25134, 31909, 40378, 50954, 64068, 80171, 100089, 124506, 154465, 191043, 235636, 289816, 355673, 435285, 531486, 647478

Offset: 0

Vaclav Kotesovec, Dec 29 2016

nonn

Convolution of A279368 and A000041.

In general, if m >= 0 and g.f. = Product_{k>=1} (1 + m*x^(k^2)) / (1-x^k), then a(n) ~ exp(Pi*sqrt((2*n)/3) + 3^(1/4)*c*n^(1/4)/ 2^(3/4) - 3*c^2/(32*Pi)) / (4*sqrt(3)*sqrt(m+1)*n), where c = -PolyLog(3/2, -m).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A033461, A279368, A280204, A280224.

nmax=50; CoefficientList[Series[Product[(1+3*x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt((2*n)/3) + 3^(1/4)*c*n^(1/4)/ 2^(3/4) - 3*c^2/(32*Pi)) / (8*sqrt(3)*n), where c = -PolyLog(3/2, -3).

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

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A279360 Expansion of Product_{k>=1} (1+2*x^(k^2)).

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A280225 G.f.: Product_{k>=1} (1 + 3*x^(k^2)) / (1-x^k).

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