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%I A116866 #13 Aug 29 2019 17:33:36
%S A116866 1,1,1,4,4,1,25,25,7,1,190,190,55,10,1,1606,1606,472,94,13,1,14506,
%T A116866 14506,4300,898,142,16,1,137089,137089,40861,8785,1495,199,19,1,
%U A116866 1338790,1338790,400567,87826,15655
%N A116866 Generalized Catalan triangle of Riordan type, called C(1,3).
%C A116866 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).
%C A116866 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.
%C A116866 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).
%C A116866 For general Riordan convolution triangles (lower triangular matrices) see the Shapiro et al. reference given in A053121.
%H A116866 Wolfdieter Lang, <a href="/A116866/a116866.txt">First 10 rows.</a>
%F A116866 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).
%e A116866 [1];[1,1];[4,4,1];[25,25,7,1];[190,190,55,10,1];...
%e A116866 Production matrix begins:
%e A116866 1, 1
%e A116866 3, 3, 1
%e A116866 9, 9, 3, 1
%e A116866 27, 27, 9, 3, 1
%e A116866 81, 81, 27, 9, 3, 1
%e A116866 243, 243, 81, 27, 9, 3, 1
%e A116866 ... _Philippe Deléham_, Sep 22 2014
%Y A116866 Row sums give A116867.
%Y A116866 Compare with the row reversed and scaled triangle A116868 (called Y(1, 3)).
%Y A116866 Cf. A115193 (similar sequence C(1,2)).
%K A116866 nonn,easy,tabl
%O A116866 0,4
%A A116866 _Wolfdieter Lang_, Mar 24 2006

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