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A156273 a(n) = 9^n*Catalan(n). 5
1, 9, 162, 3645, 91854, 2480058, 70150212, 2051893701, 61556811030, 1883638417518, 58564030799196, 1844766970174674, 58748732742485772, 1888352123865614100, 61182608813245896840, 1996082612532147384405, 65518476340761072970470, 2162109719245115408025510 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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
Column k=9 of A290605.
Sequence in context: A157553 A202438 A237024 * A051232 A362900 A157574
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Feb 07 2009
STATUS
approved

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Last modified June 24 23:28 EDT 2024. Contains 373691 sequences. (Running on oeis4.)