A339367 Number of partitions of n into an odd number of distinct squares.

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 0, 2, 1, 0, 0, 0, 2, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 1, 2, 3

Offset: 0

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

• nonn

Examples

			a(49) = 2 because we have [49] and [36, 9, 4].
		

Links

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

Crossrefs

Cf. A000290, A033461, A067659, A276516, A339364, A339365, A339366.

Programs

Mathematica

nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] - Product[(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

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

a(n) = (A033461(n) - A276516(n)) / 2.