A303906 Expansion of Product_{k>=2} 1/(1 - x^(k*(k+1)/2)).

1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 3, 1, 0, 4, 2, 0, 5, 2, 1, 7, 3, 1, 8, 4, 2, 10, 6, 2, 13, 8, 3, 15, 10, 4, 20, 12, 6, 22, 16, 8, 28, 19, 10, 33, 25, 12, 40, 29, 16, 48, 36, 19, 55, 44, 26, 65, 53, 30, 76, 64, 38, 88, 75, 46, 106, 88, 56, 119, 105, 68, 141, 122, 80, 160

Offset: 0

Author

Ilya Gutkovskiy, May 02 2018

Keywords

• nonn

Comments

First differences of A007294.

Number of partitions of n into triangular numbers > 1.

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000217, A002865, A007294, A078134, A302835.

Programs

Mathematica

nmax = 75; CoefficientList[Series[Product[1/(1 - x^(k (k + 1)/2)), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[1 + Sum[x^(j (j + 1)/2)/Product[(1 - x^(k (k + 1)/2)), {k, 2, j}], {j, 2, nmax}], {x, 0, nmax}], x]

Formula

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

a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2)^(5/3) / (2^(9/2) * sqrt(3) * Pi^(2/3) * n^(13/6)). - Vaclav Kotesovec, May 04 2018