A333153 - OEIS

%I #6 Mar 09 2020 10:48:47

%S 0,0,1,1,1,1,3,3,5,5,7,7,12,12,17,20,25,28,36,39,51,57,69,79,98,108,

%T 131,148,175,196,235,260,307,344,400,450,522,581,671,751,859,957,1097,

%U 1218,1385,1543,1744,1940,2193,2428,2735,3033,3400,3763,4215,4654

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

%H Vaclav Kotesovec, <a href="/A333153/b333153.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 * 3^(1/4) * 5^(1/2) * phi^(1/2)) = 0.07923971705837122678006319599762... 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 * 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 * x^(n*(n+1)) / Product[1 - x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

%Y Cf. A003106, A268188, A333154.

%K nonn

%O 0,7

%A _Vaclav Kotesovec_, Mar 09 2020