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%I #13 Mar 11 2020 09:50:03
%S 1,0,1,1,0,0,1,1,1,1,0,0,1,1,1,2,1,1,1,0,1,1,1,2,2,2,2,2,1,1,2,1,1,2,
%T 2,3,3,3,3,3,3,2,3,2,2,3,2,3,4,4,4,5,5,5,5,4,5,5,4,4,4,4,5,6,5,6,7,7,
%U 8,8,8,8,9,8,8,8,7,8,8,8,8,9,9,10,11,11
%N G.f.: Sum_{k>=0} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)).
%H Vaclav Kotesovec, <a href="/A333179/b333179.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.3207396095989103757477946185... = sqrt((1 - (2/(23*(23 + 3*sqrt(69))))^(1/3) + ((1/2)*(23 + 3*sqrt(69)))^(1/3)/23^(2/3))/3)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 8*s - 23*s^2 + 23*s^3 = 0.
%F Limit_{n->infinity} A306734(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...
%t nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%t nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]
%Y Cf. A003106, A053261, A306734.
%K nonn
%O 0,16
%A _Vaclav Kotesovec_, Mar 10 2020
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