A114499 Triangle read by rows: number of Dyck paths of semilength n having k 3-bridges of a given shape (0<=k<=floor(n/3)). A 3-bridge is a subpath of the form UUUDDD or UUDUDD starting at level 0.

1, 1, 2, 4, 1, 12, 2, 37, 5, 119, 12, 1, 390, 36, 3, 1307, 114, 9, 4460, 376, 25, 1, 15452, 1262, 78, 4, 54207, 4310, 255, 14, 192170, 14934, 863, 44, 1, 687386, 52397, 2967, 145, 5, 2477810, 185780, 10338, 492, 20, 8992007, 664631, 36424, 1712, 70, 1

Offset: 0

Author

Emeric Deutsch, Dec 04 2005

Keywords

• nonn
• tabf

Comments

Row n has 1+floor(n/3) terms. Row sums are the Catalan numbers (A000108). Column 0 is A114500. Sum(kT(n,k),k=0..floor(n/3))=Catalan(n-2) (n>=3; A000108).

Examples

			T(4,1)=2 because we have UD(UUUDDD) and (UUUDDD)UD (or UD(UUDUDD) and (UUDUDD)UD). The 3-bridges are shown between parentheses.
Triangle starts:
1;
1;
2;
4,1;
12,2;
37,5;
119,12,1;
390,36,3;
1307,114,9;
		

Crossrefs

Cf. A000108, A114500.

Programs

Maple

C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3-t*z^3): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 17 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form

Formula

G.f.=1/(1+z^3-tz^3-zC), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.