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%I #29 May 28 2018 16:28:57
%S 1,1,3,9,19,46,100,218,460,965,1975,3993,7975,15712,30650,59150,
%T 113093,214300,402812,751165,1390714,2557004,4670770,8479232,15302657,
%U 27462424,49021252,87057783,153850769,270614429,473850031,826125184,1434286323,2480145226
%N Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).
%C Convolved with aerated A000294: [1, 0, 2, 0, 4, 0, 10, 0, 26, ...] = A000294. - _Gary W. Adamson_, Jun 13 2009
%C This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(n+1)/2, g(n) = -1. - _Seiichi Manyama_, Nov 14 2017
%H Seiichi Manyama, <a href="/A028377/b028377.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F a(n) ~ 7^(1/8) * exp(2 * 7^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) + 3^(3/2) * 5^(1/2) * Zeta(3) * n^(1/2) / (2 * 7^(1/2) * Pi^2) - 3^(13/4) * 5^(5/4) * Zeta(3)^2 * n^(1/4) / (4 * 7^(5/4) * Pi^5) + 2025 * Zeta(3)^3 / (98*Pi^8)) / (2^(49/24) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117. - _Vaclav Kotesovec_, Mar 11 2015
%F a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(d+1)*(-1)^(1+n/d). - _Seiichi Manyama_, Nov 14 2017
%F G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^3)). - _Ilya Gutkovskiy_, May 28 2018
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(i*(i+1)/2, j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 03 2013
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[i*(i+1)/2, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Oct 13 2014, after _Alois P. Heinz_ *)
%Y Cf. A000294. - _Gary W. Adamson_, Jun 13 2009
%Y Cf. A027999, A258341, A258342, A258343, A258344, A258345, A258346.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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