A156128 a(n) = 6^n * Catalan(n).

1, 6, 72, 1080, 18144, 326592, 6158592, 120092544, 2401850880, 48997757952, 1015589892096, 21327387734016, 452796847276032, 9702789584486400, 209580255024906240, 4558370546791710720, 99747873141559787520, 2194453209114315325440, 48508965675158549299200

Offset

0,2

Comments

Number of Dyck n-paths with two types of up step and three types of down step. - David Scambler, Jun 21 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = 6^n * A000108(n).

From Gary W. Adamson, Jul 18 2011: (Start)

a(n) is the upper left term in M^n, M = an infinite square production matrix:

6, 6, 0, 0, 0, 0, ...

6, 6, 6, 0, 0, 0, ...

6, 6, 6, 6, 0, 0, ...

6, 6, 6, 6, 6, 0, ...

... (End)

E.g.f.: KummerM(1/2, 2, 24*x). - Peter Luschny, Aug 26 2012

G.f.: c(6*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013

a(n) = Sum{k=0..n} A085880(n,k) * 5^k. - Philippe Deléham, Nov 15 2013

G.f.: 1/(1 - 6*x/(1 - 6*x/(1 - 6*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017

Sum_{n>=0} 1/a(n) = 588/529 + 864*arctan(1/sqrt(23)) / (529*sqrt(23)). - Vaclav Kotesovec, Nov 23 2021

Sum_{n>=0} (-1)^n/a(n) = 564/625 - 432*log(3/2) / 3125. - Amiram Eldar, Jan 25 2022

D-finite with recurrence (n+1)*a(n) +12*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

Maple

A156128_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 6*(a[w-1]+add(a[j]*a[w-j-1], j=1..w-1)) od; convert(a, list)end: A156128_list(16); # Peter Luschny, May 19 2011

Mathematica

Table[CatalanNumber[n]6^n, {n, 0, 16}] (* Alonso del Arte, Jul 19 2011 *)

Prog

(Magma) [6^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Crossrefs

Cf. A000108, A005159, A085880, A151374, A151403, A156058.

Column k=6 of A290605.

Sequence in context: A303342 A332705 A237021 * A052678 A052719 A196882

Adjacent sequences: A156125 A156126 A156127 * A156129 A156130 A156131

Keyword

easy,nonn

Author

Philippe Deléham, Feb 04 2009

Status

approved