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a(n) = 5^n * Catalan(n).
11

%I #52 Sep 08 2022 08:45:41

%S 1,5,50,625,8750,131250,2062500,33515625,558593750,9496093750,

%T 164023437500,2870410156250,50784179687500,906860351562500,

%U 16323486328125000,295863189697265625,5395152282714843750,98911125183105468750,1822047042846679687500

%N a(n) = 5^n * Catalan(n).

%C From _Joerg Arndt_, Oct 22 2012: (Start)

%C Number of strings of length 2*n of five different types of balanced parentheses.

%C The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)

%C Number of Dyck paths of length 2n in which the step U=(1,1) come in 5 colors. [_José Luis Ramírez Ramírez_, Jan 31 2013]

%H Vincenzo Librandi, <a href="/A156058/b156058.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = 5^n*A000108(n).

%F From _Gary W. Adamson_, Jul 18 2011: (Start)

%F a(n) is the upper left term in M^n, M = the infinite square production matrix:

%F 5, 5, 0, 0, 0, 0,...

%F 5, 5, 5, 0, 0, 0,...

%F 5, 5, 5, 5, 0, 0,...

%F 5, 5, 5, 5, 5, 0,...

%F ... (End)

%F E.g.f.: KummerM(1/2, 2, 20*x). - _Peter Luschny_, Aug 26 2012

%F D-finite with recurrence (n+1)*a(n) -10*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Oct 06 2012

%F G.f.: c(5*x) with c(x) the o.g.f. of A000108 (Catalan). - _Philippe Deléham_, Nov 15 2013

%F a(n) = Sum_{k=0..n} A085880(n,k)*4^k. - _Philippe Deléham_, Nov 15 2013

%F G.f.: 1/(1 - 5*x/(1 - 5*x/(1 - 5*x/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Apr 19 2017

%F Sum_{n>=0} 1/a(n) = 410/361 + 600*arctan(1/sqrt(19)) / (361*sqrt(19)). - _Vaclav Kotesovec_, Nov 23 2021

%F Sum_{n>=0} (-1)^n/a(n) = 130/147 - 200*arctanh(1/sqrt(21)) / (147*sqrt(21)). - _Amiram Eldar_, Jan 25 2022

%p A156058_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

%p for w from 1 to n do a[w] := 5*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od; convert(a,list)end: A156058_list(16); # _Peter Luschny_, May 19 2011

%p A156058 := proc(n)

%p 5^n*A000108(n) ;

%p end proc: # _R. J. Mathar_, Oct 06 2012

%t Table[5^n CatalanNumber[n],{n,0,20}] (* _Harvey P. Dale_, Mar 13 2011 *)

%o (Magma) [5^n*Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 19 2011

%Y Cf. A000108, A005159, A085880, A151374, A151403.

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Feb 03 2009