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
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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)
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