A112698 Partial sum of (Catalan numbers A000108 multiplied by powers of 4).

1, 5, 37, 357, 3941, 46949, 587621, 7616357, 101332837, 1375876965, 18987759461, 265554114405, 3755416368997, 53610591434597, 771525112379237, 11181285666076517, 163041321978836837, 2390321854565988197

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0,..,n} C(k)*4^k, n>=0, with C(n):=A000108(n).

G.f.: c(4*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) = (17*n-7)*a(n-1) - 8*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 16^(n+1)/(15*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

Mathematica

CoefficientList[Series[(1-Sqrt[1-16*x])/(8*x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
With[{nn=20}, Accumulate[4^Range[0, nn] CatalanNumber[Range[0, nn]]]] (* Harvey P. Dale, Mar 11 2023 *)

Prog

(PARI) x='x+O('x^50); Vec((1-sqrt(1-16*x))/(8*x*(1-x))) \\ G. C. Greubel, Mar 17 2017

Crossrefs

Fifth column (m=4) of triangle A112705.

Sequence in context: A004208 A198077 A208813 * A234953 A344051 A025168

Adjacent sequences: A112695 A112696 A112697 * A112699 A112700 A112701

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 31 2005

Extensions

Definition clarified by Harvey P. Dale, Mar 11 2023

Status

approved