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A116873 Generalized Catalan numbers C(2,3;n)=C(3,2;n). 5
1, 1, 5, 49, 653, 10201, 174965, 3188641, 60623645, 1189050025, 23884139525, 488869387249, 10159948737581, 213822249696121, 4547793322315157, 97600834975487809, 2110916340429978173 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This sequence appears in the Derrida et al. 1992 reference as Z_{N}=:Y_{N}(N+1), N >=0, for alpha =2, beta = 3 (or alpha=3, beta=2). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1. See also the Liggett reference, proposition 3.19, p. 269, with lambda for alpha and rho for 1-beta.

REFERENCES

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

G. Schuetz and E. Domany, Phase Transitions in an Exactly Soluble one-Dimensional Exclusion Process, J. Stat. Phys. 72 (1993) 277-295, eq. (2.18), p. 283, with eqs. (2.13)-(2.15).

T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.

LINKS

Table of n, a(n) for n=0..16.

J. Abate, W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 4.

FORMULA

G.f.: (1+5*x+12*x^2*c(6*x))/((1+3*x)*(1+x)) with the g.f. c(x) for A000108 (Catalan numbers).

a(n)=((-1)^(n+1))*(3^n-2)+6*sum(((-1)^k)*C(n-2-k)*6^(n-2-k)*(3^(k+1)-1),k=0..n-2), with C(n):=A000108(n) (Catalan numbers) and the sum is replaced by 0 for n=0,1. Proof from the g.f. after partial fraction decomposition. [Wolfdieter Lang, May 05 2006]

(1-n)*a(n) +4*(5*n-14)*a(n-1) +3*(31*n-79)*a(n-2) +36*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 06 2013

CROSSREFS

Sequence in context: A243945 A297513 A228511 * A324361 A089914 A267220

Adjacent sequences:  A116870 A116871 A116872 * A116874 A116875 A116876

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Mar 24 2006

STATUS

approved

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Last modified January 21 22:54 EST 2021. Contains 340359 sequences. (Running on oeis4.)