A339375 Number of partitions of n into an even number of distinct triangular numbers.

1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 0, 3, 1, 0, 2, 1, 1, 1, 3, 1, 2, 0, 2, 2, 0, 2, 3, 3, 1, 2, 2, 2, 2, 2, 1, 4, 4, 1, 3, 2, 3, 2, 3, 1, 5, 4, 2, 4, 2, 4, 4, 3, 2, 6, 4, 3, 4, 5, 2, 3, 6, 5, 6, 5, 4, 5, 5, 4, 5, 6, 4

Offset: 0

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

• nonn

Examples

			a(31) = 3 because we have [28, 3], [21, 10] and [21, 6, 3, 1].
		

Links

• Index entries for sequences related to partitions
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000217, A024940, A067661, A292518, A339366, A339373, A339374, A339376.

Programs

Mathematica

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

Formula

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

a(n) = (A024940(n) + A292518(n)) / 2.