A281615 Expansion of Sum_{i>=1} x^(i*(i+1)/2)/(1 - x^(i*(i+1)/2)) / Product_{j>=1} (1 - x^(j*(j+1)/2)).

1, 2, 4, 6, 8, 13, 17, 21, 30, 37, 44, 60, 72, 83, 107, 127, 144, 181, 210, 236, 289, 333, 371, 446, 507, 562, 664, 750, 825, 965, 1083, 1187, 1371, 1530, 1668, 1912, 2122, 2307, 2618, 2896, 3138, 3540, 3897, 4211, 4717, 5180, 5581, 6222, 6803, 7317, 8116, 8853, 9497, 10486, 11401, 12215, 13430, 14572, 15576, 17067

Offset: 1

Author

Ilya Gutkovskiy, Jan 25 2017

Keywords

• nonn

Comments

Total number of parts in all partitions of n into nonzero triangular numbers (A000217).

Convolution of A007294 and A007862.

Examples

			a(6) = 13 because we have [6], [3, 3], [3, 1, 1, 1], [1, 1, 1, 1, 1, 1] and 1 + 2 + 4 + 6 = 13.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000217, A007294, A007862, A281541.

Programs

Mathematica

nmax = 60; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2)/(1 - x^(i (i + 1)/2)), {i, 1, nmax}]/Product[1 - x^(j (j + 1)/2), {j, 1, nmax}], {x, 0, nmax}], x]]

Formula

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

a(n) ~ exp(3*zeta(3/2)^(2/3) * (Pi*n)^(1/3)/2) * zeta(3/2)^(1/3) / (2^(3/2) * sqrt(3) * Pi^(4/3) * n^(5/6)). - Vaclav Kotesovec, Sep 15 2021