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A116880 Generalized Catalan triangle, called CM(1,2).

%I A116880 #22 Aug 05 2025 10:02:08
%S A116880 1,1,3,3,7,13,13,29,41,67,67,147,195,247,381,381,829,1069,1277,1545,
%T A116880 2307,2307,4995,6339,7379,8451,9975,14589,14589,31485,39549,45373,
%U A116880 50733,56829,66057,95235,95235,205059,255747,290691,320707,351187,388099,446455,636925
%N A116880 Generalized Catalan triangle, called CM(1,2).
%C A116880 This triangle generalizes the 'new' Catalan triangle A028364 (which could be called CM(1,1); M stands for author Meeussen).
%H A116880 Nathaniel Johnston, <a href="/A116880/b116880.txt">Table of n, a(n) for n = 0..2500</a>
%H A116880 Wolfdieter Lang, <a href="/A116880/a116880.txt">First 10 rows</a>.
%F A116880 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).
%e A116880 Triangle begins:
%e A116880      1;
%e A116880      1,    3;
%e A116880      3,    7,   13;
%e A116880     13,   29,   41,   67;
%e A116880     67,  147,  195,  247,  381;
%e A116880    381,  829, 1069, 1277, 1545, 2307;
%e A116880   2307, 4995, 6339, 7379, 8451, 9975, 14589;
%p A116880 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 A116880 Column m=0 gives A064062.
%Y A116880 Row sums give A116881.
%K A116880 nonn,easy,tabl
%O A116880 0,3
%A A116880 _Wolfdieter Lang_, Mar 24 2006