A156270 a(n) = 8^n*Catalan(n).

1, 8, 128, 2560, 57344, 1376256, 34603008, 899678208, 23991418880, 652566593536, 18034567675904, 504967894925312, 14294475794808832, 408413594137395200, 11762311511156981760, 341107033823552471040, 9952299339793060331520

Offset

0,2

Comments

A quarter of the count of And/Or-Trees with 2 variables [Chauvin]. - R. J. Mathar, Apr 01 2012

Links

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

Brigitte Chauvin, Philippe Flajolet, Daniele Gardy and Bernhard Gittenberger, And/Or Tree Revisited, Combinat., Probal. Comput., Vol. 13, No. 4-5 (2004), pp. 475-497.

Formula

a(n) = 8^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:

8, 8, 0, 0, 0, 0, ...

8, 8, 8, 0, 0, 0, ...

8, 8, 8, 8, 0, 0, ...

8, 8, 8, 8, 8, 0, ...

... (End)

E.g.f.: KummerM(1/2, 2, 32*x). - Peter Luschny, Aug 26 2012

G.f.: c(8*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)*7^k. - Philippe Deléham, Nov 15 2013

G.f.: 1/(1 - 8*x/(1 - 8*x/(1 - 8*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017

(n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Apr 14 2018

Sum_{n>=0} 1/a(n) = 1040/961 + 1536*arctan(1/sqrt(31)) / (961*sqrt(31)). - Vaclav Kotesovec, Nov 23 2021

Sum_{n>=0} (-1)^n/a(n) = 112/121 - 512*arctanh(1/sqrt(33)) / (363*sqrt(33)). - Amiram Eldar, Jan 25 2022

D-finite with recurrence +(n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

Mathematica

Table[8^n*CatalanNumber[n], {n, 0, 20}] (* Wesley Ivan Hurt, Dec 28 2013 *)

Prog

(Magma) [8^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Crossrefs

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266.

Column k=8 of A290605.

Sequence in context: A183497 A370102 A237023 * A051189 A371299 A113135

Adjacent sequences: A156267 A156268 A156269 * A156271 A156272 A156273

Keyword

nonn,easy

Author

Philippe Deléham, Feb 07 2009

Status

approved