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%I #10 Apr 14 2018 14:34:39
%S 1,2,3,5,7,9,13,17,21,27,34,41,51,62,73,88,105,122,144,168,193,225,
%T 260,296,340,388,438,498,564,632,713,802,894,1001,1118,1239,1380,1533,
%U 1692,1873,2070,2275,2508,2760,3022,3317,3637,3969,4341,4742,5159,5624,6125,6645,7220,7839
%N Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k*(k+1)/2)).
%C Partial sums of A007294.
%C Number of partitions of n into triangular numbers if there are two kinds of 1's.
%H Alois P. Heinz, <a href="/A302835/b302835.txt">Table of n, a(n) for n = 0..20000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: (1/(1 - x))*Sum_{j>=0} x^(j*(j+1)/2)/Product_{k=1..j} (1 - x^(k*(k+1)/2)).
%F From _Vaclav Kotesovec_, Apr 13 2018: (Start)
%F 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)).
%F a(n) ~ 2 * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A007294(n). (End)
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
%p b(n, i-1)+(t->`if`(t>n, 0, b(n-t, i)))(i*(i+1)/2))
%p end:
%p a:= n-> b(n, isqrt(2*n)):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Apr 13 2018
%t nmax = 55; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
%t 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]
%Y Cf. A000070, A000217, A007294, A298435, A302833.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Apr 13 2018
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