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%I #14 Jun 18 2019 04:02:35
%S 1,1,1,3,3,7,13,21,33,67,115,183,333,541,937,1635,2643,4327,7573,
%T 12069,20025,33427,54259,87375,144669,231541,374809,607443,970539,
%U 1545367,2502205,3947541,6270057,9997867,15776083,24832503,39351309,61552501,96632689
%N G.f.: Sum_{n>=0} n! * x^(n*(n+1)/2) / Product_{k=1..n} (1 - k*x^k).
%H Seiichi Manyama, <a href="/A204858/b204858.txt">Table of n, a(n) for n = 0..6263</a>
%F G.f.: 1/(1 - x/(1 - 2*x^2*(1-x)/(1 - 3*x^3*(1-2*x^2)/(1 - 4*x^4*(1-3*x^3)/(1 - 5*x^5*(1-4*x^4)/(1 - 6*x^6*(1-5*x^5)/(1 -...)))))), a continued fraction.
%F From _Vaclav Kotesovec_, Jun 18 2019: (Start)
%F a(n) ~ c * 3^(n/3), where
%F c = 8007.60951343849770902289074154120578227939552369... if mod(n,3)=0
%F c = 8007.30566699919825273673656299755925992856381905... if mod(n,3)=1
%F c = 8007.19663204881021378993302255541874790731157021... if mod(n,3)=2
%F (End)
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 21*x^7 +...
%e where A(x) = 1 + x/(1-x) + 2!*x^3/((1-x)*(1-2*x^2)) + 3!*x^6/((1-x)*(1-2*x^2)*(1-3*x^3)) + 4!*x^10/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...
%t Table[SeriesCoefficient[Sum[n!*x^Binomial[n + 1, 2]/Product[(1 - k*x^k), {k, 1, n}], {n, 0, 100}], {x, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Dec 19 2017 *)
%o (PARI) {a(n)=polcoeff(1+sum(m=1,n,m!*x^(m*(m+1)/2)/prod(k=1,m,1-k*x^k+x*O(x^n))),n)}
%Y Cf. A032020, A204857, A204856.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jan 20 2012
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