A067323 Catalan triangle A028364 with row reversion.

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, 11934, 10504, 9646, 8986, 8398, 7810, 7150, 6292, 4862

Offset: 0

Author

Wolfdieter Lang, Feb 05 2002

Keywords

• nonn
• easy
• tabl

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 A000108(n)%20(Catalan)%20in%20this%20reference.%20See%20also%20the%20Derrida%20et%20al.%201993%20reference.%20In%20the%20Liggett%201999%20reference%20mu">{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 _{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.

Examples

			Triangle begins:
     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;
  ...
		

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).

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 14.
• Wolfdieter Lang, First 10 rows.

Crossrefs

Cf. A001700 (row sums).

Cf. A039598, A039599.

T(2n,n) gives A201205.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
    end:
T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(b((n+1)$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 28 2015

Mathematica

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 *)

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). - Philippe Deléham, Feb 18 2004

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