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

1, 1, 3, 3, 7, 13, 13, 29, 41, 67, 67, 147, 195, 247, 381, 381, 829, 1069, 1277, 1545, 2307, 2307, 4995, 6339, 7379, 8451, 9975, 14589, 14589, 31485, 39549, 45373, 50733, 56829, 66057, 95235, 95235, 205059, 255747, 290691, 320707, 351187, 388099, 446455, 636925

Offset: 0

Author

Wolfdieter Lang, Mar 24 2006

Keywords

• nonn
• easy
• tabl

Comments

This triangle generalizes the 'new' Catalan triangle A028364 (which could be called CM(1,1); M stands for author Meeussen).

Examples

			Triangle begins:
     1;
     1,    3;
     3,    7,   13;
    13,   29,   41,   67;
    67,  147,  195,  247,  381;
   381,  829, 1069, 1277, 1545, 2307;
  2307, 4995, 6339, 7379, 8451, 9975, 14589;
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..2500
• Wolfdieter Lang, First 10 rows.

Crossrefs

Column m=0 gives A064062.

Row sums give A116881.

Programs

Maple

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

Formula

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).