cp's OEIS Frontend

A156058 a(n) = 5^n * Catalan(n).

%I A156058 #52 Sep 08 2022 08:45:41
%S A156058 1,5,50,625,8750,131250,2062500,33515625,558593750,9496093750,
%T A156058 164023437500,2870410156250,50784179687500,906860351562500,
%U A156058 16323486328125000,295863189697265625,5395152282714843750,98911125183105468750,1822047042846679687500
%N A156058 a(n) = 5^n * Catalan(n).
%C A156058 From _Joerg Arndt_, Oct 22 2012: (Start)
%C A156058 Number of strings of length 2*n of five different types of balanced parentheses.
%C A156058 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 A156058 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 A156058 Vincenzo Librandi, <a href="/A156058/b156058.txt">Table of n, a(n) for n = 0..200</a>
%F A156058 a(n) = 5^n*A000108(n).
%F A156058 From _Gary W. Adamson_, Jul 18 2011: (Start)
%F A156058 a(n) is the upper left term in M^n, M = the infinite square production matrix:
%F A156058   5, 5, 0, 0, 0, 0,...
%F A156058   5, 5, 5, 0, 0, 0,...
%F A156058   5, 5, 5, 5, 0, 0,...
%F A156058   5, 5, 5, 5, 5, 0,...
%F A156058   ... (End)
%F A156058 E.g.f.: KummerM(1/2, 2, 20*x). - _Peter Luschny_, Aug 26 2012
%F A156058 D-finite with recurrence (n+1)*a(n) -10*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Oct 06 2012
%F A156058 G.f.: c(5*x) with c(x) the o.g.f. of A000108 (Catalan). - _Philippe Deléham_, Nov 15 2013
%F A156058 a(n) = Sum_{k=0..n} A085880(n,k)*4^k. - _Philippe Deléham_, Nov 15 2013
%F A156058 G.f.: 1/(1 - 5*x/(1 - 5*x/(1 - 5*x/(1 - ...)))), a continued fraction. - _Ilya Gutkovskiy_, Apr 19 2017
%F A156058 Sum_{n>=0} 1/a(n) = 410/361 + 600*arctan(1/sqrt(19)) / (361*sqrt(19)). - _Vaclav Kotesovec_, Nov 23 2021
%F A156058 Sum_{n>=0} (-1)^n/a(n) = 130/147 - 200*arctanh(1/sqrt(21)) / (147*sqrt(21)). - _Amiram Eldar_, Jan 25 2022
%p A156058 A156058_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A156058 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 A156058 := proc(n)
%p A156058     5^n*A000108(n) ;
%p A156058 end proc: # _R. J. Mathar_, Oct 06 2012
%t A156058 Table[5^n CatalanNumber[n],{n,0,20}]  (* _Harvey P. Dale_, Mar 13 2011 *)
%o A156058 (Magma) [5^n*Catalan(n): n in [0..20]]; // _Vincenzo Librandi_, Jul 19 2011
%Y A156058 Cf. A000108, A005159, A085880, A151374, A151403.
%K A156058 nonn,easy
%O A156058 0,2
%A A156058 _Philippe Deléham_, Feb 03 2009