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A333141 G.f.: Sum_{k>=1} (k^2 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

%I A333141 #11 Mar 20 2020 07:42:47
%S A333141 0,1,1,1,5,5,9,9,13,22,26,35,48,57,70,88,117,135,173,207,261,304,374,
%T A333141 433,528,628,739,864,1032,1198,1416,1639,1914,2212,2569,2949,3433,
%U A333141 3920,4511,5150,5925,6732,7720,8736,9969,11284,12823,14444,16395,18457,20836
%N A333141 G.f.: Sum_{k>=1} (k^2 * x^(k^2) / Product_{j=1..k} (1 - x^j)).
%H A333141 Vaclav Kotesovec, <a href="/A333141/b333141.txt">Table of n, a(n) for n = 0..10000</a>
%F A333141 a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(1/4), where c = A333155^2 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.076061100391958657489521534823556... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
%p A333141 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A333141       `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A333141     end:
%p A333141 a:= n-> add(k^2 * b(n-k^2, k), k=1..floor(sqrt(n))):
%p A333141 seq(a(n), n=0..50); # after _Alois P. Heinz_
%t A333141 nmax = 50; CoefficientList[Series[Sum[n^2 * x^(n^2) / Product[1 - x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%Y A333141 Cf. A003114, A268188.
%K A333141 nonn
%O A333141 0,5
%A A333141 _Vaclav Kotesovec_, Mar 09 2020