|
| |
|
|
A115193
|
|
Generalized Catalan triangle of Riordan type, called C(1,2).
|
|
8
| |
|
|
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
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
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(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.
|
|
|
LINKS
| Nathaniel Johnston, Table of n, a(n) for n = 0..5150 (up to row 100)
W. 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).
|
|
|
EXAMPLE
| Triangle begins:
1
1 1
3 3 1
13 13 5 1
67 67 27 7 1
...
|
|
|
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
| Row sums give A115197. Compare with the row reversed and scaled triangle A115195.
Sequence in context: A143603 A094021 A062746 * A039797 A143171 A112292
Adjacent sequences: A115190 A115191 A115192 * A115194 A115195 A115196
|
|
|
KEYWORD
| nonn,easy,tabl
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 23 2006
|
| |
|
|
