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A333151 G.f.: Sum_{k>=1} (k^3 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

%I A333151 #6 Mar 09 2020 10:47:13
%S A333151 0,1,1,1,9,9,17,17,25,52,60,87,122,149,184,238,337,391,517,635,825,
%T A333151 970,1224,1433,1778,2176,2585,3074,3736,4414,5292,6223,7354,8626,
%U A333151 10135,11785,13915,16068,18701,21600,25141,28884,33512,38288,44165,50494,57961
%N A333151 G.f.: Sum_{k>=1} (k^3 * x^(k^2) / Product_{j=1..k} (1 - x^j)).
%H A333151 Vaclav Kotesovec, <a href="/A333151/b333151.txt">Table of n, a(n) for n = 0..10000</a>
%F A333151 a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(3/4), where c = A333155^3 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.04512265567211918167849606290245... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
%p A333151 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A333151       `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A333151     end:
%p A333151 a:= n-> add(k^3 * b(n-k^2, k), k=1..floor(sqrt(n))):
%p A333151 seq(a(n), n=0..50);  # after _Alois P. Heinz_
%t A333151 nmax = 50; CoefficientList[Series[Sum[n^3*x^(n^2)/Product[1-x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%Y A333151 Cf. A003114, A268188, A333141.
%K A333151 nonn
%O A333151 0,5
%A A333151 _Vaclav Kotesovec_, Mar 09 2020