A339364 Number of partitions of n into an even number of squares.

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 3, 3, 3, 3, 4, 4, 6, 4, 7, 6, 7, 7, 8, 9, 11, 9, 13, 12, 14, 14, 16, 16, 20, 17, 23, 22, 25, 25, 28, 29, 33, 31, 37, 38, 41, 42, 45, 48, 54, 51, 61, 60, 67, 67, 72, 76, 84, 81, 93, 93, 102, 104, 110, 117, 125, 125, 139, 140, 153

Offset: 0

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

• nonn

Examples

			a(10) = 3 because we have [9, 1], [4, 4, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
		

Links

• Index entries for sequences related to partitions
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000290, A001156, A027187, A292520, A339365, A339366, A339367.

Programs

Mathematica

nmax = 70; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] + Product[1/(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

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

a(n) = (A001156(n) + A292520(n)) / 2.