A302835 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k*(k+1)/2)).

1, 2, 3, 5, 7, 9, 13, 17, 21, 27, 34, 41, 51, 62, 73, 88, 105, 122, 144, 168, 193, 225, 260, 296, 340, 388, 438, 498, 564, 632, 713, 802, 894, 1001, 1118, 1239, 1380, 1533, 1692, 1873, 2070, 2275, 2508, 2760, 3022, 3317, 3637, 3969, 4341, 4742, 5159, 5624, 6125, 6645, 7220, 7839

Offset: 0

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

• nonn

Comments

Partial sums of A007294.

Number of partitions of n into triangular numbers if there are two kinds of 1's.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000
• Index entries for sequences related to partitions

Crossrefs

Cf. A000070, A000217, A007294, A298435, A302833.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+(t->`if`(t>n, 0, b(n-t, i)))(i*(i+1)/2))
    end:
a:= n-> b(n, isqrt(2*n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 13 2018

Mathematica

nmax = 55; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[1/(1 - x) Sum[x^(j (j + 1)/2)/Product[(1 - x^(k (k + 1)/2)), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x))*Sum_{j>=0} x^(j*(j+1)/2)/Product_{k=1..j} (1 - x^(k*(k+1)/2)).

From Vaclav Kotesovec, Apr 13 2018: (Start)

a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2)^(1/3) / (2^(5/2) * sqrt(3) * Pi^(4/3) * n^(5/6)).

a(n) ~ 2 * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A007294(n). (End)