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A067323 Catalan triangle A028364 with row reversion.

%I A067323 #31 Dec 27 2024 08:55:59
%S A067323 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 A067323 227,202,174,132,1430,1001,869,785,715,645,561,429,4862,3432,3003,
%U A067323 2739,2529,2333,2123,1859,1430,16796,11934,10504,9646,8986,8398,7810,7150,6292,4862
%N A067323 Catalan triangle A028364 with row reversion.
%C A067323 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 A067323 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 A067323 The first column sequences (diagonals of A028364) are: A000108(n+1), A000245, A067324-6 for m=0..4.
%D A067323 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 A067323 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 A067323 T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, pp. 269, 275.
%D A067323 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 A067323 Alois P. Heinz, <a href="/A067323/b067323.txt">Rows n = 0..140, flattened</a>
%H A067323 Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, <a href="https://arxiv.org/abs/2412.07873">Lucky cars and lucky spots in parking functions</a>, arXiv:2412.07873 [math.CO], 2024. See p. 14.
%H A067323 Wolfdieter Lang, <a href="/A067323/a067323.txt">First 10 rows</a>.
%F A067323 a(n,m) = A028364(n,n-m), n>=m>=0, else 0.
%F A067323 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 A067323 T(n,k) = Sum_{j>=0} A039598(n-k,j)*A039599(k,j). - _Philippe Deléham_, Feb 18 2004
%F A067323 G.f. for diagonal sequences: see g.f. for columns of A028364.
%e A067323 Triangle begins:
%e A067323      1;
%e A067323      2,    1;
%e A067323      5,    3,    2;
%e A067323     14,    9,    7,    5;
%e A067323     42,   28,   23,   19,   14;
%e A067323    132,   90,   76,   66,   56,   42;
%e A067323    429,  297,  255,  227,  202,  174,  132;
%e A067323   1430, 1001,  869,  785,  715,  645,  561,  429;
%e A067323   4862, 3432, 3003, 2739, 2529, 2333, 2123, 1859, 1430;
%e A067323   ...
%p A067323 b:= proc(n, i) option remember; `if`(n=0, 1, add(
%p A067323       expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
%p A067323     end:
%p A067323 T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(b((n+1)$2)):
%p A067323 seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 28 2015
%t A067323 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 A067323 Cf. A001700 (row sums).
%Y A067323 Cf. A039598, A039599.
%Y A067323 T(2n,n) gives A201205.
%K A067323 nonn,easy,tabl
%O A067323 0,2
%A A067323 _Wolfdieter Lang_, Feb 05 2002