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A064094
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Triangle composed of generalized Catalan numbers.
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23
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 13, 4, 1, 1, 1, 42, 67, 25, 5, 1, 1, 1, 132, 381, 190, 41, 6, 1, 1, 1, 429, 2307, 1606, 413, 61, 7, 1, 1, 1, 1430, 14589, 14506, 4641, 766, 85, 8, 1, 1, 1, 4862, 95235
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OFFSET
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0,8
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COMMENTS
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The column m sequence (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=Y_{N}(N+1), N >=0, for alpha = m, beta = 1 (or alpha = 1, beta = m). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1.
The column sequences (without leading zeros) are: A000012, A000108, A064062-3, A064087-93 for m=0..10, respectively. Row sums give A064095.
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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.
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LINKS
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Table of n, a(n) for n=0..57.
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FORMULA
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G.f. for column m: (x^m)/(1-x*c(m*x))= (x^m)*((m-1)+m*x*c(m*x))/(m-1+x) with the g.f. c(x) of Catalan numbers A000108.
a(n, m)= sum((n-m-k)*binomial(n-m-1+k, k)*(m^k)/(n-m), k=0..n-m-1) = ((1/(1-m))^(n-m)*(1-m*sum(C(k)*(m*(1-m))^k, k=0..n-m-1)), n-m >= 1; a(n, n)=1; a(n, m)=0 if n<m; with C(k)=A000108(k) (Catalan).
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CROSSREFS
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Diagonals : A000012, A000012, A000027, A001844, A064096, A064302, A064303, A064304, A064305.
Sequence in context: A069739 A066060 A008550 * A090182 A111673 A121391
Adjacent sequences: A064091 A064092 A064093 * A064095 A064096 A064097
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang, Sep 13 2001
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STATUS
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approved
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