

A114499


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


2



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
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OFFSET

0,3


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(n2) (n>=3; A000108).


LINKS

Table of n, a(n) for n=0..50.


FORMULA

G.f.=1/(1+z^3tz^3zC), where C=[1sqrt(14z)]/(2z) is the Catalan function.


EXAMPLE

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


MAPLE

C:=(1sqrt(14*z))/2/z: G:=1/(1z*C+z^3t*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


CROSSREFS

Cf. A000108, A114500.
Sequence in context: A246188 A135333 A124503 * A030730 A117131 A204117
Adjacent sequences: A114496 A114497 A114498 * A114500 A114501 A114502


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Dec 04 2005


STATUS

approved



