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

Offset

0,3

Comments

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

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..2500

W. Lang: First 10 rows.

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).

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

Crossrefs

Row sums give A116881.

Sequence in context: A088859 A177942 A360873 * A051123 A096188 A187873

Adjacent sequences: A116877 A116878 A116879 * A116881 A116882 A116883

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Mar 24 2006

Status

approved