A064343 Generalized Catalan numbers C(5,5; n).

1, 1, 10, 325, 16750, 1056250, 74237500, 5580578125, 439118593750, 35714849218750, 2978473867187500, 253316015488281250, 21887247402929687500, 1915840314586914062500, 169529844641289062500000

Offset: 0

Author

Wolfdieter Lang, Oct 12 2001

Keywords

• nonn
• easy

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

• J. Abate, W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, corollary 6.

Crossrefs

Cf. A000108, A064342.

Formula

a(n) = ((25^(n-1))/(n-1))*Sum_{m=0..n-2} (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/5)^(m+1)), n >= 2, a(0) := 1 =: a(1).

G.f.: (1-9*x*c(25*x))/(1-5*x*c(25*x))^2 = c(25*x)*(9+16*c(25*x))/(1+4*c(25*x))^2 = (9*c(25*x)*(5*x)^2+8*(2+7*x))/(4+5*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.

4*(-n+1)*a(n) +5*(79*n-200)*a(n-1) +250*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017