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%I #8 Mar 09 2020 12:50:02
%S 0,0,1,1,1,1,5,5,9,9,13,13,26,26,39,48,61,70,92,101,139,157,195,229,
%T 292,326,405,464,559,634,779,870,1047,1188,1406,1604,1888,2127,2493,
%U 2823,3271,3683,4283,4802,5525,6221,7112,7992,9137,10210,11625,13013,14734
%N G.f.: Sum_{k>=1} (k^2 * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)).
%C In general, if m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)), then a(n) ~ r^m * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)), where r = A333155 = sqrt(15) * log(phi) / Pi = 0.59324221500336912718413761733... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
%H Vaclav Kotesovec, <a href="/A333154/b333154.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(1/4), where c = A333155^2 / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)) = 0.04700834526394839955207674000683... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p end:
%p a:= n-> add(k^2 * b(n-k*(k+1), k), k=1..floor(sqrt(n))):
%p seq(a(n), n=0..60); # after _Alois P. Heinz_
%t nmax = 60; CoefficientList[Series[Sum[n^2 * x^(n*(n+1)) / Product[1 - x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%Y Cf. A003106, A268188, A333153.
%K nonn
%O 0,7
%A _Vaclav Kotesovec_, Mar 09 2020
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