A300417 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^2.

1, 2, 1, 2, 4, 2, 3, 6, 3, 4, 10, 8, 5, 10, 11, 8, 14, 16, 11, 18, 22, 18, 23, 22, 22, 34, 31, 26, 39, 40, 33, 50, 56, 36, 53, 74, 51, 62, 86, 68, 77, 98, 86, 88, 102, 106, 120, 130, 120, 136, 157, 134, 157, 194, 155, 182, 241, 194, 196, 256, 237, 236, 288, 282, 273, 324

Offset: 0

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

• nonn

Comments

Number of partitions of n into distinct triangular parts (A000217), with 2 types of each part.

Self-convolution of A024940.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000217, A022567, A024940, A279226, A298435.

Programs

Mathematica

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

Formula

G.f.: Product_{k>=1} (1 + x^A000217(k))^2.

a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2)/2)^(2/3) * n^(1/3)) * ((sqrt(2)-1) * Zeta(3/2) / (2*Pi))^(1/3) / (4*sqrt(3) * n^(5/6)). - Vaclav Kotesovec, Mar 05 2018