A116866 Generalized Catalan triangle of Riordan type, called C(1,3).

1, 1, 1, 4, 4, 1, 25, 25, 7, 1, 190, 190, 55, 10, 1, 1606, 1606, 472, 94, 13, 1, 14506, 14506, 4300, 898, 142, 16, 1, 137089, 137089, 40861, 8785, 1495, 199, 19, 1, 1338790, 1338790, 400567, 87826, 15655

Offset: 0

Author

Wolfdieter Lang, Mar 24 2006

Keywords

• nonn
• easy
• tabl

Comments

This triangle is the second 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(a(n,m)*x^n,m=0..n) is D(x,z)=g(z)/(1 - x*z*c(3*z))= g(z)*(3*z-x*z*(1-3*z*c(3*z)))/(3*z-x*z+(x*z)^2), with g(z) and c(z) defined below.

This is the Riordan triangle named (g(x),x*c(3*x)) with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers). g(x) is the o.g.f. of A064063 (C(3;n) Catalan generalization).

For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.

Examples

			[1];[1,1];[4,4,1];[25,25,7,1];[190,190,55,10,1];...
Production matrix begins:
1, 1
3, 3, 1
9, 9, 3, 1
27, 27, 9, 3, 1
81, 81, 27, 9, 3, 1
243, 243, 81, 27, 9, 3, 1
... _Philippe Deléham_, Sep 22 2014
		

Links

• Wolfdieter Lang, First 10 rows.

Crossrefs

Row sums give A116867.

Compare with the row reversed and scaled triangle A116868 (called Y(1, 3)).

Cf. A115193 (similar sequence C(1,2)).

Formula

G.f. for column m>=0 is g(x)*(x*c(3*x))^m, with g(x):=(1+3*x*c(3*x)/2)/(1+x/2) and c(x) is the o.g.f. of A000108 (Catalan numbers).