A067324 -id:A067324 - cp's OEIS frontend

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

Wolfdieter Lang, Feb 05 2002

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.

			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;
  ...

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).

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.

Cf. A001700 (row sums).
Cf. A039598, A039599.
T(2n,n) gives A201205.

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

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 *)

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.

A287548 Triangle read by rows: T(n,k), where each row begins with the Catalan number for n nonintersecting arches and transitions through k generations of eliminating and reducing arch configurations to an end row entry equal to number of semi-meander solutions for n arches.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 14, 9, 7, 4, 42, 28, 23, 16, 10, 132, 90, 76, 57, 42, 24, 429, 297, 255, 199, 156, 108, 66, 1430, 1001, 869, 695, 563, 420, 304, 174, 4862, 3432, 3003, 2442, 2019, 1568, 1210, 836, 504

Offset: 1

Roger Ford, May 26 2017

			Triangle begins:
n\k    1    2    3    4    5    6    7    8
1:     1
2:     2    1
3:     5    3    2
4:     14   9    7    4
5:     42   28   23   16   10
6:     132  90   76   57   42   24
7:     429  297  255  199  156  108  66
8:     1430 1001 869  695  563  420  304  174
...
Capital letters (U,D) represent beginning and end of first and last arch. Only 1 UD ends arch sequence in next generation.
Reduction of arches:            Elimination of arches:
(middle D U = new arch U D in the next arch generation)
            /\
     /\    //\\                      /\/\/\/\  = UDududUD
    //\\/\///\\\  = UudDudUuuddD        /\
        /\  /\                         /  \
     /\//\\//\\   =  UDuuddUudD       //\/\\   =  UududD
                                        end
For n=3 C(n)=5  nonintersecting arch configurations:
   UuuddD   UududD   UudDUD   UDUudD   UDudUD   T(3,1)=5
    end      end      UDUD     UDUD     UudD    T(3,2)=3
                       UD       UD       end    T(3,3)=2

Cf. A000108, A000245, A067324, A000682.

T(n,1) = Catalan Numbers C(n)= A000108(n).
Conjectured:
T(n,2) = C(n) - C(n-1) = A000245(n-1).
T(n,3) = C(n) - C(n-1) - C(n-2) = A067324(n-3).
T(n,4) = C(n) - C(n-1) - 2*C(n-2) - C(n-3).
T(n,n) = semi-meander solutions = A000682(n-1).

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A067323 Catalan triangle A028364 with row reversion.

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A067325 Fourth column of triangle A067323.

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A287548 Triangle read by rows: T(n,k), where each row begins with the Catalan number for n nonintersecting arches and transitions through k generations of eliminating and reducing arch configurations to an end row entry equal to number of semi-meander solutions for n arches.

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