A333153 G.f.: Sum_{k>=1} (k * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)).

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, 131, 148, 175, 196, 235, 260, 307, 344, 400, 450, 522, 581, 671, 751, 859, 957, 1097, 1218, 1385, 1543, 1744, 1940, 2193, 2428, 2735, 3033, 3400, 3763, 4215, 4654

Offset: 0

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003106, A268188, A333154.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k * b(n-k*(k+1), k), k=1..floor(sqrt(n))):
seq(a(n), n=0..60);  # after Alois P. Heinz

Mathematica

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]

Formula

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.