A127153 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k UDUD's starting at level 0; here U=(1,1), D=(1,-1) (0<=k<=n-1).

1, 1, 1, 1, 4, 0, 1, 11, 2, 0, 1, 33, 6, 2, 0, 1, 105, 17, 7, 2, 0, 1, 343, 56, 19, 8, 2, 0, 1, 1148, 185, 64, 21, 9, 2, 0, 1, 3916, 624, 214, 72, 23, 10, 2, 0, 1, 13563, 2144, 726, 244, 80, 25, 11, 2, 0, 1, 47571, 7468, 2510, 832, 275, 88, 27, 12, 2, 0, 1, 168625, 26317

Offset

0,5

Comments

Row 0 has one entry; row n has n entries (n>=1). Row sums yield the Catalan numbers (A000108). Column 0 yields A127154. The reference does not list the 0's (p. 2920, lines 3,4).

Links

Table of n, a(n) for n=0..68.

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Formula

G.f.: (1+z-t*z)/(1+z-t*z+z^2-t*z^2-z*C(1+z-t*z)), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan g.f. (see A000108).

Or, g.f.: (1+(1-t)*z)*C/(1+(1-t)*z*(1+z*C)).

Example

T(4,1)=2 because we have UDUDUUDD and UUDDUDUD; T(4,3)=1 because we have UDUDUDUD.
Triangle starts:
1;
1;
1,1;
4,0,1;
11,2,0,1;
33,6,2,0,1;
105,17,7,2,0,1;

Maple

G:=(1+z-t*z)/(1+z-t*z+z^2-t*z^2-z*C*(1+z-t*z)): C:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G, z=0, 16)): for n from 0 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: 1; for n from 1 to 13 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form

Crossrefs

Cf. A000108, A127154.

Sequence in context: A121301 A059056 A344393 * A178979 A228270 A335748

Adjacent sequences: A127150 A127151 A127152 * A127154 A127155 A127156

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 27 2007

Extensions

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

Status

approved