A339365 Number of partitions of n into an odd number of squares.

0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 2, 3, 3, 3, 4, 5, 4, 6, 5, 7, 7, 7, 8, 10, 9, 12, 10, 14, 13, 14, 15, 18, 17, 21, 20, 24, 24, 25, 27, 31, 30, 35, 34, 40, 40, 42, 45, 50, 50, 56, 55, 64, 65, 68, 72, 78, 79, 88, 85, 99, 99, 105, 110, 118, 122, 131, 132, 146, 149

Offset

0,10

Links

Table of n, a(n) for n=0..70.

Index entries for sequences related to partitions

Index entries for sequences related to sums of squares

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.

Example

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

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]

Crossrefs

Cf. A000290, A001156, A027193, A292520, A339364, A339366, A339367.

Sequence in context: A247856 A059660 A194003 * A035456 A035664 A095246

Adjacent sequences: A339362 A339363 A339364 * A339366 A339367 A339368

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 01 2020

Status

approved