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

Author

Wolfdieter Lang, Feb 23 2006

Keywords

• nonn
• easy
• tabl

Comments

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.

Examples

			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
		

Links

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

Crossrefs

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

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

Programs

Maple

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

Mathematica

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 *)

Formula

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