OFFSET
0,4
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.
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (up to row 100)
Wolfdieter Lang, First 10 rows.
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
EXAMPLE
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
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
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Feb 23 2006
STATUS
approved