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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.)