A333154 G.f.: Sum_{k>=1} (k^2 * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)).

0, 0, 1, 1, 1, 1, 5, 5, 9, 9, 13, 13, 26, 26, 39, 48, 61, 70, 92, 101, 139, 157, 195, 229, 292, 326, 405, 464, 559, 634, 779, 870, 1047, 1188, 1406, 1604, 1888, 2127, 2493, 2823, 3271, 3683, 4283, 4802, 5525, 6221, 7112, 7992, 9137, 10210, 11625, 13013, 14734

Offset: 0

Author

Vaclav Kotesovec, Mar 09 2020

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

Cf. A003106, A268188, A333153.

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^2 * 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^2 * 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 / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)) = 0.04700834526394839955207674000683... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.