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A067323
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Catalan triangle A028364 with row reversion.
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10
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1, 2, 1, 5, 3, 2, 14, 9, 7, 5, 42, 28, 23, 19, 14, 132, 90, 76, 66, 56, 42, 429, 297, 255, 227, 202, 174, 132, 1430, 1001, 869, 785, 715, 645, 561, 429, 4862, 3432, 3003, 2739, 2529, 2333, 2123, 1859, 1430, 16796
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(N,p) equals X_{N}(N+1,p) := T_{N,p} for alpha= 1 =beta and N>=p>=1 in the Derrida et al. 1992 reference. The one-point correlation functions <tau_{K}>_{N} for alpha= 1 =beta equal a(N,K)/C(N+1) with C(n)=A000108(n) (Catalan) in this reference. See also the Derrida et al. 1993 reference. In the Liggett 1999 reference mu_{N}{eta:eta(k)=1} of prop. 3.38, p. 275 is identical with <tau_{k}>_{N} and rho=0 and lambda=1.
Identity for each row n>=1: a(n,m)+a(n,n-m+1)= C(n+1), with C(n+1)=A000108(n+1)(Catalan) for every m=1..floor((n+1)/2). E.g. a(2k+1,k+1)=C(2*(k+1)).
The first column sequences (diagonals of A028364) are: A000108(n+1), A000245, A067324-6 for m=0..4.
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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. (19) - (23), 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; eqs. (43), (44), pp. 1501-2 and eq.(81) with eqs.(80) and (81).
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, pp. 269, 275.
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).
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LINKS
| W. Lang: First 10 rows.
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FORMULA
| a(n, m)= A028364(n, n-m), n>=m>=0, else 0.
G.f. for column m>=1 (without leading zeros): (c(x)^3)sum(C(m-1, k)*c(x)^k, k=0..m-1), with C(n, m) := (m+1)*binomial(2*n-m, n-m)/(n+1) (Catalan convolutions A033184); and for m=0: c^2(x), where c(x) is g.f. of A000108 (Catalan).
T(n, k) = Sum_{j>=0} A039598(n-k, j)*A039599(k, j). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 18 2004
G.f. for diagonal sequences: see g.f. for columns of A028364.
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EXAMPLE
| {1}; {2,1}; {5,3,2}; {14,9,7,5}; ...; n=3: 14 = 9+5 = 7+7.
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CROSSREFS
| Cf. A001700 (row sums).
Cf. A039598, A039599.
Sequence in context: A171177 A171176 A067418 * A106534 A123346 A163840
Adjacent sequences: A067320 A067321 A067322 * A067324 A067325 A067326
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 5 2002
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