A333152 - OEIS

%I #8 Mar 09 2020 12:49:58

%S 0,1,1,1,17,17,33,33,49,130,146,227,324,405,502,664,1017,1179,1613,

%T 2031,2721,3220,4166,4921,6204,7840,9379,11352,14028,16882,20520,

%U 24511,29286,34864,41401,48741,58417,68144,80207,93698,110325,128124,150436,173424

%N G.f.: Sum_{k>=1} (k^4 * x^(k^2) / Product_{j=1..k} (1 - x^j)).

%C In general, if m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k^2) / Product_{j=1..k} (1 - x^j)), then a(n) ~ r^m * phi^(1/2) * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(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="/A333152/b333152.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ c * exp(2*Pi*sqrt(n/15)) * n^(5/4), where c = A333155^4 * phi^(1/2) / (2 * 3^(1/4) * 5^(1/2)) = 0.026768664197762321048783840410317... 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^4 * b(n-k^2, k), k=1..floor(sqrt(n))):

%p seq(a(n), n=0..50);  # after _Alois P. Heinz_

%t nmax = 50; CoefficientList[Series[Sum[n^4*x^(n^2)/Product[1-x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

%Y Cf. A003114, A268188, A333141, A333151.

%K nonn

%O 0,5

%A _Vaclav Kotesovec_, Mar 09 2020