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%I #14 Aug 29 2019 16:21:00
%S 1,1,3,3,7,13,13,29,41,67,67,147,195,247,381,381,829,1069,1277,1545,
%T 2307,2307,4995,6339,7379,8451,9975,14589,14589,31485,39549,45373,
%U 50733,56829,66057,95235,95235,205059,255747,290691,320707,351187,388099,446455
%N Generalized Catalan triangle, called CM(1,2).
%C This triangle generalizes the 'new' Catalan triangle A028364 (which could be called CM(1,1); M stands for author Meeussen).
%H Nathaniel Johnston, <a href="/A116880/b116880.txt">Table of n, a(n) for n = 0..2500</a>
%H W. Lang: <a href="/A116880/a116880.txt">First 10 rows.</a>
%F G.f. for columns m >= 0 (without leading zeros): c(2;x)*Sum_{k=0..m} C(1,2;m,k)*(2*c(2*x))^k with c(2;x):=(1+2*x*c(2*x))/(1+x) the g.f. of A064062 and c(x) is the g.f. of A000108 (Catalan). C(1,2;n,m) is the triangle A115193(n,m).
%p lim:=8: 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: gf2:=g*sum(a[s,k]*(2*c)^k,k=0..s): for s from 0 to lim do t:=taylor(gf2, x, lim+1): for n from 0 to lim do b[n,s]:=coeff(t, x, n):od:od: seq(seq(b[n-s,s],s=0..n),n=0..lim); # _Nathaniel Johnston_, Apr 30 2011
%Y Row sums give A116881.
%K nonn,easy,tabl
%O 0,3
%A _Wolfdieter Lang_, Mar 24 2006
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