A115194 -id:A115194 - cp's OEIS frontend

A115193 Generalized Catalan triangle of Riordan type, called C(1,2).

1, 1, 1, 3, 3, 1, 13, 13, 5, 1, 67, 67, 27, 7, 1, 381, 381, 157, 45, 9, 1, 2307, 2307, 963, 291, 67, 11, 1, 14589, 14589, 6141, 1917, 477, 93, 13, 1, 95235, 95235, 40323, 12867, 3363, 723, 123, 15, 1, 636925

Offset: 0

Wolfdieter Lang, Feb 23 2006

This triangle is the first of a family of generalizations of the Catalan convolution triangle A033184 (which belongs to the Bell subgroup of the Riordan group).
The o.g.f. of the row polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^n is D(x,z) = g(z)/(1 - x*z*c(2*z)) = g(z)*(2*z-x*z*(1-2*z*c(2*z)))/(2*z-x*z+(x*z)^2), with g(z) and c(z) defined below.
This is the Riordan triangle named (g(x),x*c(2*x)) with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064062 (C(2;n) Catalan generalization).
The column sequences (without leading zeros) are A064062, A064062(n+1), A084076, A115194, A115202-A115204, for m=0..6.
For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

			Triangle begins:
   1;
   1,  1;
   3,  3,  1;
  13, 13,  5,  1;
  67, 67, 27,  7,  1;
  ...
Production matrix begins:
    1,   1;
    2,   2,   1;
    4,   4,   2,   1;
    8,   8,   4,   2,   1;
   16,  16,   8,   4,   2,   1;
   32,  32,  16,   8,   4,   2,   1;
   64,  64,  32,  16,   8,   4,   2,   1;
  128, 128,  64,  32,  16,   8,   4,   2,   1;
  ... _Philippe Deléham_, Sep 22 2014

Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (up to row 100)
Wolfdieter Lang, First 10 rows.

Row sums give A115197. Compare with the row reversed and scaled triangle A115195.

Cf. A116866 (similar sequence C(1,3)).

lim:=7: c:=(1-sqrt(1-8*x))/(4*x): g:=(1+2*x*c)/(1+x): gf1:=g*(x*c)^m: for m from 0 to lim do t:=taylor(gf1,x,lim+1): for n from 0 to lim do a[n,m]:=coeff(t,x,n):od:od: seq(seq(a[n,m],m=0..n),n=0..lim); # Nathaniel Johnston, Apr 30 2011

A110510[n_, k_] := (k/n)*Binomial[2*n - k - 1, n - k]*2^(n - k);
T[n_, k_] := If[n == 0, 1, Sum[A110510[n, i], {i, k, n}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 17 2025 *)

G.f. for column m>=0 is g(x)*(x*c(2*x))^m, with g(x):=(1+2*x*c(2*x))/(1+x) and c(x) is the o.g.f. of A000108 (Catalan numbers).

T(n,k) = Sum_{i=k..n} A110510(n,i) for 0 <= k <= n. - Werner Schulte, Mar 24 2019

A115195 Triangle of numbers, called Y(1,2), related to generalized Catalan numbers A062992(n) = C(2;n+1) = A064062(n+1).

Original entry on oeis.org

1, 2, 3, 4, 10, 13, 8, 28, 54, 67, 16, 72, 180, 314, 381, 32, 176, 536, 1164, 1926, 2307, 64, 416, 1488, 3816, 7668, 12282, 14589, 128, 960, 3936, 11568, 26904, 51468, 80646, 95235, 256, 2176, 10048, 33184, 86992, 189928, 351220, 541690, 636925, 512, 4864

Offset: 0

Wolfdieter Lang, Feb 23 2006

This triangle Y(1,2) appears in the totally asymmetric exclusion process for the (unphysical) values alpha=1, beta=2. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
The main diagonal (M=1) gives the generalized Catalan sequence C(2,n+1):=A064062(n+1).
The diagonal sequences give A064062(n+1), 2*A084076, 4*A115194, 8*A115202, 16*A115203, 32*A115204 for n+1>= M=1,..,6.

			Triangle begins:
   1;
   2,  3;
   4, 10,  13;
   8, 28,  54,  67;
  16, 72, 180, 314, 381;
  ...

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. (20), (21), p. 672.
Wolfdieter Lang, First 10 rows.

Row sums give A084076.

G.f. m-th diagonal, m>=1: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^m)/(2*x*(1+x)) with c(x) the o.g.f. of A000108 (Catalan).

cp's OEIS Frontend

A115193 Generalized Catalan triangle of Riordan type, called C(1,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula