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

%I A156128 #50 Sep 08 2022 08:45:41
%S A156128 1,6,72,1080,18144,326592,6158592,120092544,2401850880,48997757952,
%T A156128 1015589892096,21327387734016,452796847276032,9702789584486400,
%U A156128 209580255024906240,4558370546791710720,99747873141559787520,2194453209114315325440,48508965675158549299200
%N A156128 a(n) = 6^n * Catalan(n).
%C A156128 Number of Dyck n-paths with two types of up step and three types of down step. - _David Scambler_, Jun 21 2013
%H A156128 Vincenzo Librandi, <a href="/A156128/b156128.txt">Table of n, a(n) for n = 0..200</a>
%F A156128 a(n) = 6^n * A000108(n).
%F A156128 From _Gary W. Adamson_, Jul 18 2011: (Start)
%F A156128 a(n) is the upper left term in M^n, M = an infinite square production matrix:
%F A156128   6, 6, 0, 0, 0, 0, ...
%F A156128   6, 6, 6, 0, 0, 0, ...
%F A156128   6, 6, 6, 6, 0, 0, ...
%F A156128   6, 6, 6, 6, 6, 0, ...
%F A156128   ... (End)
%F A156128 E.g.f.: KummerM(1/2, 2, 24*x). - _Peter Luschny_, Aug 26 2012
%F A156128 G.f.: c(6*x) with c(x) the o.g.f. of A000108 (Catalan). - _Philippe Deléham_, Nov 15 2013
%F A156128 a(n) = Sum{k=0..n} A085880(n,k) * 5^k. - _Philippe Deléham_, Nov 15 2013
%F A156128 G.f.: 1/(1 - 6*x/(1 - 6*x/(1 - 6*x/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Aug 08 2017
%F A156128 Sum_{n>=0} 1/a(n) = 588/529 + 864*arctan(1/sqrt(23)) / (529*sqrt(23)). - _Vaclav Kotesovec_, Nov 23 2021
%F A156128 Sum_{n>=0} (-1)^n/a(n) = 564/625 - 432*log(3/2) / 3125. - _Amiram Eldar_, Jan 25 2022
%F A156128 D-finite with recurrence (n+1)*a(n) +12*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Mar 21 2022
%p A156128 A156128_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A156128 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
%t A156128 Table[CatalanNumber[n]6^n, {n, 0, 16}] (* _Alonso del Arte_, Jul 19 2011 *)
%o A156128 (Magma) [6^n*Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 19 2011
%Y A156128 Cf. A000108, A005159, A085880, A151374, A151403, A156058.
%Y A156128 Column k=6 of A290605.
%K A156128 easy,nonn
%O A156128 0,2
%A A156128 _Philippe Deléham_, Feb 04 2009