%I #9 Mar 11 2020 09:49:25
%S 0,1,1,0,2,2,2,2,0,3,3,3,6,3,3,3,4,4,4,8,8,8,8,8,4,9,9,5,10,10,15,15,
%T 15,15,15,15,16,16,11,17,17,18,24,24,24,30,30,30,30,31,31,31,32,26,33,
%U 34,41,41,42,49,49,56,56,56,64,64,57,65,58,59,67,68
%N G.f.: Sum_{k>=1} (k * x^(k^2) * Product_{j=1..k} (1 + x^j)).
%H Vaclav Kotesovec, <a href="/A333180/b333180.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) ~ c * A333198^sqrt(n), where c = 0.3836313809149103736315...
%F Limit_{n->infinity} a(n) / A333181(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[n*x^(n^2)*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%t nmax = 100; p = 1; s = 0; Do[p = Expand[p*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += k*p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]
%Y Cf. A268188, A306734, A333181.
%K nonn
%O 0,5
%A _Vaclav Kotesovec_, Mar 10 2020