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

1, 5, 50, 625, 8750, 131250, 2062500, 33515625, 558593750, 9496093750, 164023437500, 2870410156250, 50784179687500, 906860351562500, 16323486328125000, 295863189697265625, 5395152282714843750, 98911125183105468750, 1822047042846679687500

Offset: 0

Author

Philippe Deléham, Feb 03 2009

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

Magma

[5^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

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 *)

Formula

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

From Gary W. Adamson, Jul 18 2011: (Start)

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

5, 5, 0, 0, 0, 0,...

5, 5, 5, 0, 0, 0,...

5, 5, 5, 5, 0, 0,...

5, 5, 5, 5, 5, 0,...

... (End)

E.g.f.: KummerM(1/2, 2, 20*x). - Peter Luschny, Aug 26 2012

D-finite with recurrence (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

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

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