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Catalan triangle A028364 with row reversion.
12

%I #27 Aug 28 2019 16:22:16

%S 1,2,1,5,3,2,14,9,7,5,42,28,23,19,14,132,90,76,66,56,42,429,297,255,

%T 227,202,174,132,1430,1001,869,785,715,645,561,429,4862,3432,3003,

%U 2739,2529,2333,2123,1859,1430,16796,11934,10504,9646,8986,8398,7810,7150,6292,4862

%N Catalan triangle A028364 with row reversion.

%C 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.

%C 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)).

%C The first column sequences (diagonals of A028364) are: A000108(n+1), A000245, A067324-6 for m=0..4.

%D 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.

%D 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).

%D T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, pp. 269, 275.

%D 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).

%H Alois P. Heinz, <a href="/A067323/b067323.txt">Rows n = 0..140, flattened</a>

%H W. Lang: <a href="/A067323/a067323.txt">First 10 rows.</a>

%F a(n,m) = A028364(n,n-m), n>=m>=0, else 0.

%F 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).

%F T(n,k) = Sum_{j>=0} A039598(n-k,j)*A039599(k,j). - _Philippe Deléham_, Feb 18 2004

%F G.f. for diagonal sequences: see g.f. for columns of A028364.

%e Triangle begins:

%e 1;

%e 2, 1;

%e 5, 3, 2;

%e 14, 9, 7, 5;

%e 42, 28, 23, 19, 14;

%e 132, 90, 76, 66, 56, 42;

%e 429, 297, 255, 227, 202, 174, 132;

%e 1430, 1001, 869, 785, 715, 645, 561, 429;

%e 4862, 3432, 3003, 2739, 2529, 2333, 2123, 1859, 1430;

%e ...

%p b:= proc(n, i) option remember; `if`(n=0, 1, add(

%p expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(b((n+1)$2)):

%p seq(T(n), n=0..10); # _Alois P. Heinz_, Nov 28 2015

%t t[n_, k_] := Sum[ CatalanNumber[n - j]*CatalanNumber[j], {j, 0, k}]; Flatten[ Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}]] (* _Jean-François Alcover_, Jul 17 2013 *)

%Y Cf. A001700 (row sums).

%Y Cf. A039598, A039599.

%Y T(2n,n) gives A201205.

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Feb 05 2002