A302833 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)).

1, 2, 3, 4, 6, 8, 10, 12, 15, 19, 23, 27, 32, 38, 44, 50, 58, 67, 77, 87, 99, 112, 126, 140, 156, 175, 195, 216, 239, 265, 292, 320, 351, 385, 422, 460, 503, 549, 598, 648, 703, 763, 826, 892, 963, 1041, 1122, 1206, 1296, 1394, 1498, 1605, 1721, 1845, 1977, 2112, 2256, 2410, 2573

Offset

0,2

Comments

Partial sums of A001156.

Number of partitions of n into squares if there are two kinds of 1's.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Index entries for sequences related to partitions

Index entries for sequences related to sums of squares

Formula

G.f.: (1/(1 - x))*Sum_{j>=0} x^(j^2)/Product_{k=1..j} (1 - x^(k^2)).

From Vaclav Kotesovec, Apr 13 2018: (Start)

a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2*Pi^(3/2) * sqrt(3*n)).

a(n) ~ 2^(4/3) * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A001156(n). (End)

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 13 2018

Mathematica

nmax = 58; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 58; CoefficientList[Series[1/(1 - x) Sum[x^j^2/Product[(1 - x^k^2), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000070, A000290, A001156, A078134, A279225, A302835.

Sequence in context: A139542 A347656 A238616 * A093717 A359754 A330899

Adjacent sequences: A302830 A302831 A302832 * A302834 A302835 A302836

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 13 2018

Status

approved