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A116873
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Generalized Catalan numbers C(2,3;n)=C(3,2;n).
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5
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1, 1, 5, 49, 653, 10201, 174965, 3188641, 60623645, 1189050025, 23884139525, 488869387249, 10159948737581, 213822249696121, 4547793322315157, 97600834975487809, 2110916340429978173
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OFFSET
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0,3
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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