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

0,4

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.

Links

Table of n, a(n) for n=0..40.

Wolfdieter Lang, First 10 rows.

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).

Example

[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

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)).

Sequence in context: A371898 A079507 A098364 * A126280 A170986 A071207

Adjacent sequences: A116863 A116864 A116865 * A116867 A116868 A116869

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Mar 24 2006

Status

approved