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

0,2

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.

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

W. Lang: First 10 rows.

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.

Example

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;
  ...

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 *)

Crossrefs

Cf. A001700 (row sums).

Cf. A039598, A039599.

T(2n,n) gives A201205.

Sequence in context: A213849 A067418 A287548 * A106534 A123346 A163840

Adjacent sequences: A067320 A067321 A067322 * A067324 A067325 A067326

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Feb 05 2002

Status

approved