A339374 Number of partitions of n into an odd number of triangular numbers.

0, 1, 0, 2, 0, 2, 1, 3, 1, 4, 3, 4, 4, 6, 5, 9, 7, 10, 9, 14, 10, 19, 15, 21, 18, 27, 22, 34, 30, 37, 37, 47, 43, 57, 56, 64, 66, 80, 75, 96, 94, 108, 110, 131, 125, 155, 154, 173, 178, 207, 201, 240, 245, 267, 280, 315, 315, 364, 374, 406, 423, 477, 473, 543, 555, 604

Offset

0,4

Links

Table of n, a(n) for n=0..65.

Index entries for sequences related to partitions

Index to sequences related to polygonal numbers

Formula

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

a(n) = (A007294(n) - A292519(n)) / 2.

Example

a(7) = 3 because we have [3, 3, 1], [3, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1].

Mathematica

nmax = 65; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}] - Product[1/(1 + x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Crossrefs

Cf. A000217, A007294, A027193, A292519, A339365, A339373, A339375, A339376.

Sequence in context: A304034 A029183 A213423 * A265753 A138110 A080233

Adjacent sequences: A339371 A339372 A339373 * A339375 A339376 A339377

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 02 2020

Status

approved