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

0,2

Links

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

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

Mathematica

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

Prog

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

Crossrefs

Fourth column (m=3) of triangle A112705.

Cf. A000108.

Sequence in context: A052650 A198053 A197925 * A327994 A392744 A113717

Adjacent sequences: A112694 A112695 A112696 * A112698 A112699 A112700

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 31 2005

Status

approved