A156273 a(n) = 9^n*Catalan(n).

1, 9, 162, 3645, 91854, 2480058, 70150212, 2051893701, 61556811030, 1883638417518, 58564030799196, 1844766970174674, 58748732742485772, 1888352123865614100, 61182608813245896840, 1996082612532147384405, 65518476340761072970470, 2162109719245115408025510

Offset

0,2

Links

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

Formula

a(n) = 9^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:

9, 9, 0, 0, 0, 0, ...

9, 9, 9, 0, 0, 0, ...

9, 9, 9, 9, 0, 0, ...

9, 9, 9, 9, 9, 0, ...

... (End)

E.g.f.: KummerM(1/2, 2, 36*x). - Peter Luschny, Aug 26 2012

G.f.: c(9*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)*8^k. - Philippe Deléham, Nov 15 2013

G.f.: 1/(1 - 9*x/(1 - 9*x/(1 - 9*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017

Sum_{n>=0} 1/a(n) = 1314/1225 + 1944*arctan(1/sqrt(35)) / (1225*sqrt(35)). - Vaclav Kotesovec, Nov 23 2021

Sum_{n>=0} (-1)^n/a(n) = 1278/1369 - 1944*arctanh(1/sqrt(37)) / (1369*sqrt(37)). - Amiram Eldar, Jan 25 2022

D-finite with recurrence (n+1)*a(n) +18*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

Mathematica

Table[9^n CatalanNumber[n], {n, 0, 20}] (* Harvey P. Dale, Sep 09 2012 *)

Prog

(Magma)[9^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Crossrefs

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266, A156270.

Column k=9 of A290605.

Sequence in context: A157553 A202438 A237024 * A051232 A362900 A157574

Adjacent sequences: A156270 A156271 A156272 * A156274 A156275 A156276

Keyword

nonn,easy

Author

Philippe Deléham, Feb 07 2009

Status

approved