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

%I A156273 #39 Sep 08 2022 08:45:41
%S A156273 1,9,162,3645,91854,2480058,70150212,2051893701,61556811030,
%T A156273 1883638417518,58564030799196,1844766970174674,58748732742485772,
%U A156273 1888352123865614100,61182608813245896840,1996082612532147384405,65518476340761072970470,2162109719245115408025510
%N A156273 a(n) = 9^n*Catalan(n).
%H A156273 Vincenzo Librandi, <a href="/A156273/b156273.txt">Table of n, a(n) for n = 0..200</a>
%F A156273 a(n) = 9^n*A000108(n).
%F A156273 From _Gary W. Adamson_, Jul 18 2011: (Start)
%F A156273 a(n) is the upper left term in M^n, M = an infinite square production matrix:
%F A156273   9, 9, 0, 0, 0, 0, ...
%F A156273   9, 9, 9, 0, 0, 0, ...
%F A156273   9, 9, 9, 9, 0, 0, ...
%F A156273   9, 9, 9, 9, 9, 0, ...
%F A156273   ... (End)
%F A156273 E.g.f.: KummerM(1/2, 2, 36*x). - _Peter Luschny_, Aug 26 2012
%F A156273 G.f.: c(9*x) with c(x) the o.g.f. of A000108 (Catalan). - _Philippe Deléham_, Nov 15 2013
%F A156273 a(n) = Sum{k=0..n} A085880(n,k)*8^k. - _Philippe Deléham_, Nov 15 2013
%F A156273 G.f.: 1/(1 - 9*x/(1 - 9*x/(1 - 9*x/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Aug 08 2017
%F A156273 Sum_{n>=0} 1/a(n) = 1314/1225 + 1944*arctan(1/sqrt(35)) / (1225*sqrt(35)). - _Vaclav Kotesovec_, Nov 23 2021
%F A156273 Sum_{n>=0} (-1)^n/a(n) = 1278/1369 - 1944*arctanh(1/sqrt(37)) / (1369*sqrt(37)). - _Amiram Eldar_, Jan 25 2022
%F A156273 D-finite with recurrence (n+1)*a(n) +18*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Mar 21 2022
%t A156273 Table[9^n CatalanNumber[n],{n,0,20}] (* _Harvey P. Dale_, Sep 09 2012 *)
%o A156273 (Magma)[9^n*Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 19 2011
%Y A156273 Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266, A156270.
%Y A156273 Column k=9 of A290605.
%K A156273 nonn,easy
%O A156273 0,2
%A A156273 _Philippe Deléham_, Feb 07 2009