A112697 Partial sum of Catalan numbers (A000108) multiplied by powers of 3.

1, 4, 22, 157, 1291, 11497, 107725, 1045948, 10428178, 106126924, 1097913928, 11511677470, 122057782762, 1306480339462, 14098243951822, 153208673236237, 1675240428936307, 18417589741637077, 203464608460961377

Offset: 0

Author

Wolfdieter Lang, Oct 31 2005

Keywords

• nonn
• easy

Links

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

Crossrefs

Fourth column (m=3) of triangle A112705.

Cf. A000108.

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-12*x])/(6*x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

PARI

x='x+O('x^50); Vec((1-sqrt(1-12*x))/(6*x*(1-x))) \\ G. C. Greubel, Mar 17 2017

Formula

a(n) = Sum_{k=0..n} C(k)*3^k, n>=0, with C(n) = A000108(n).

G.f.: c(3*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) = (13*n-5)*a(n-1) - 6*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 12^(n+1)/(11*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012