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

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

• nonn

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

Crossrefs

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

Programs

Mathematica

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

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.