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%I #23 Dec 12 2023 17:40:29
%S 1,1,1,2,2,4,5,9,11,22,27,49,68,115,157,279,372,628,914,1457,2070,
%T 3457,4840,7753,11442,17768,25824,41315,59008,92140,137212,208524,
%U 305472,477659,691381,1058019,1575694,2370618,3491693,5359888,7796346,11799263,17583757
%N G.f.: Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1 - k*x^k).
%H Seiichi Manyama, <a href="/A204856/b204856.txt">Table of n, a(n) for n = 0..6279</a> (terms 0..250 from Vaclav Kotesovec)
%F G.f.: 1/(1 - x/(1 - x^2*(1-x)/(1-x^2 - x^3*(1-2*x^2)/(1-2*x^3 - x^4*(1-3*x^3)/(1-3*x^4 - x^5*(1-4*x^4)/(1-4*x^5 -...))))), a continued fraction.
%F From _Vaclav Kotesovec_, Jun 18 2019: (Start)
%F a(n) ~ c * 3^(n/3), where
%F c = 23.5612420584121380174441491950859168338330954540437... if mod(n,3)=0
%F c = 23.5209031427848763179214171003561794127717213180726... if mod(n,3)=1
%F c = 23.5214569018665529984420312927586688667133017590049... if mod(n,3)=2
%F (End)
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 9*x^7 + 11*x^8 +...
%e where A(x) = 1 + x/(1-x) + x^3/((1-x)*(1-2*x^2)) + x^6/((1-x)*(1-2*x^2)*(1-3*x^3)) + x^10/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...
%t Table[SeriesCoefficient[Sum[x^Binomial[n + 1, 2]/Product[(1 - k*x^k), {k, 1, n}], {x, 0, 100}], {x, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Dec 19 2017 *)
%o (PARI) {a(n)=polcoeff(1+sum(m=1,n,x^(m*(m+1)/2)/prod(k=1,m,1-k*x^k+x*O(x^n))),n)}
%Y Cf. A204855, A204858.
%Y Column sums of A367955.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jan 20 2012
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