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 A156058 a(n) = 5^n * Catalan(n). 9
 1, 5, 50, 625, 8750, 131250, 2062500, 33515625, 558593750, 9496093750, 164023437500, 2870410156250, 50784179687500, 906860351562500, 16323486328125000, 295863189697265625, 5395152282714843750, 98911125183105468750, 1822047042846679687500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Joerg Arndt, Oct 22 2012: (Start) Number of strings of length 2*n of five different types of balanced parentheses. 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) 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] LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA a(n) = 5^n*A000108(n). a(n) = upper left term in M^n, M = the infinite square production matrix as follows: 5, 5, 0, 0, 0, 0,... 5, 5, 5, 0, 0, 0,... 5, 5, 5, 5, 0, 0,... 5, 5, 5, 5, 5, 0,... ... - Gary W. Adamson, Jul 18 2011 E.g.f.: KummerM(1/2, 2, 20*x). - Peter Luschny, Aug 26 2012 (n+1)*a(n) -10*(2*n-1)*a(n-1)=0. - R. J. Mathar, Oct 06 2012 G.f.: c(5*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)*4^k. - Philippe Deléham, Nov 15 2013 G.f.: 1/(1 - 5*x/(1 - 5*x/(1 - 5*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Apr 19 2017 MAPLE A156058_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1; 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 A156058 := proc(n)     5^n*A000108(n) ; end proc: # R. J. Mathar, Oct 06 2012 MATHEMATICA Table[5^n CatalanNumber[n], {n, 0, 20}]  (* Harvey P. Dale, Mar 13 2011 *) PROG (MAGMA) [5^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011 CROSSREFS Cf. A000108, A151374, A005159, A151403. Sequence in context: A078244 A233068 A237020 * A232997 A234464 A047736 Adjacent sequences:  A156055 A156056 A156057 * A156059 A156060 A156061 KEYWORD nonn,easy AUTHOR Philippe Deléham, Feb 03 2009 STATUS approved

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Last modified April 22 10:13 EDT 2019. Contains 322330 sequences. (Running on oeis4.)