A112699 Partial sum of Catalan numbers A000108 multiplied by powers of 5.

1, 6, 56, 681, 9431, 140681, 2203181, 35718806, 594312556, 10090406306, 174113843806, 3044524000056, 53828703687556, 960689055250056, 17284175383375056, 313147365080640681, 5708299647795484431

Offset: 0

Author

Wolfdieter Lang, Oct 31 2005

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Sixth column (m=5) of triangle A112705.

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-20*x])/(10*x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

PARI

x='x+O('x^50); Vec((1-sqrt(1-20*x))/(10*x*(1-x))) \\ G. C. Greubel, Mar 17 2017

Formula

a(n) = Sum_{k=0,..,n} C(k)*5^k, n>=0, with C(n):=A000108(n).

G.f.: c(5*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.

Recurrence: (n+1)*a(n) = 3*(7*n-3)*a(n-1) - 10*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 20^(n+1)/(19*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012