A338465 Number of ways to write 2*n as an ordered sum of n nonzero triangular numbers.

1, 0, 2, 0, 6, 5, 20, 42, 70, 261, 297, 1430, 1584, 7293, 9634, 35945, 60150, 176596, 366401, 886977, 2150421, 4624410, 12205074, 25065216, 67616872, 139894305, 369551925, 793214982, 2011977414, 4517758504, 10992821055, 25669627965, 60531471286, 145112506352

Offset

0,3

Comments

Also number of ways to write n as an ordered sum of n nonnegative numbers one less than a triangular number.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Formula

a(n) = [x^(2*n)] (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^n, where theta_2() is the Jacobi theta function.

a(n) = [x^n] (Sum_{k>=0} x^(k*(k + 3)/2))^n.

Mathematica

Table[SeriesCoefficient[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^n, {x, 0, 2 n}], {n, 0, 33}]

Crossrefs

Cf. A000096, A000217, A106337, A298858, A319799.

Sequence in context: A326040 A145419 A276763 * A142354 A105110 A351884

Adjacent sequences: A338462 A338463 A338464 * A338466 A338467 A338468

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 31 2021

Status

approved