%I A115193
%S A115193 1,1,1,3,3,1,13,13,5,1,67,67,27,7,1,381,381,157,45,9,1,2307,2307,963,
%T A115193 291,67,11,1,14589,14589,6141,1917,477,93,13,1,95235,95235,40323,12867,
%U A115193 3363,723,123,15,1,636925
%N A115193 Generalized Catalan triangle of Riordan type, called C(1,2).
%C A115193 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).
%C A115193 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.
%C A115193 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).
%C A115193 The column sequences (without leading zeros) are A064062, A064062(n+1),
A084076, A115194, A115202-A115204, for m=0,..,6.
%C A115193 For general Riordan convolution triangles (lower triangular matrices)
see the Shapiro et al. reference given in A053121.
%H A115193 W. Lang: <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A115193.text">
First 10 rows.</a>
%F A115193 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).
%e A115193 [1];[1,1];[3,3,1];[13,13,5,1];[67,67,27,7,1];...
%Y A115193 Row sums give A115197.
%Y A115193 Compare with the row reversed and scaled triangle A115195.
%Y A115193 Sequence in context: A143603 A094021 A062746 this_sequence A039797 A143171
A112292
%Y A115193 Adjacent sequences: A115190 A115191 A115192 this_sequence A115194 A115195
A115196
%K A115193 nonn,easy,tabl
%O A115193 0,4
%A A115193 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de),
Feb 23 2006
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