0,4

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(a(n,m)*x^n,m=0..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.

W. Lang: First 10 rows.

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).

[1];[1,1];[3,3,1];[13,13,5,1];[67,67,27,7,1];...

Row sums give A115197.

Compare with the row reversed and scaled triangle A115195.

Sequence in context: A143603 A094021 A062746 this_sequence A039797 A143171 A112292

Adjacent sequences: A115190 A115191 A115192 this_sequence A115194 A115195 A115196

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 23 2006