A261569 Expansion of Product_{k>=1} (1 + 5*x^k).

1, 5, 5, 30, 30, 55, 180, 205, 330, 480, 1230, 1380, 2255, 3030, 4530, 8555, 10680, 15330, 21330, 29730, 39480, 67380, 81505, 116280, 153030, 210930, 270805, 370080, 534330, 675480, 900480, 1180380, 1544130, 1997280, 2597280, 3304805, 4581180, 5653080

Offset: 0

Author

Vaclav Kotesovec, Aug 24 2015

Keywords

• nonn

Comments

In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - Vaclav Kotesovec, Jan 04 2016

Links

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

Crossrefs

Cf. A000009, A032302, A032308, A261568, A291698.

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 5*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015

Mathematica

nmax = 40; CoefficientList[Series[Product[1 + 5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*5^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-5, x]/6 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/6 + log(5)^2/2 + polylog(2, -1/5) = 2.74927912606080829002558751537626864449... . - Vaclav Kotesovec, Jan 04 2016

G.f.: Sum_{i>=0} 5^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018