A128744 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the first peak equal to k (1 <= k <= n).

1, 1, 2, 3, 3, 4, 10, 10, 8, 8, 36, 36, 29, 20, 16, 137, 137, 111, 78, 48, 32, 543, 543, 442, 315, 200, 112, 64, 2219, 2219, 1813, 1306, 848, 496, 256, 128, 9285, 9285, 7609, 5527, 3649, 2200, 1200, 576, 256, 39587, 39587, 32521, 23779, 15901, 9802, 5552, 2848

Offset: 1

Author

Emeric Deutsch, Mar 31 2007

Keywords

• tabl
• nonn

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1) (down) and L=(-1,-1) (left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.

Row sums yield A002212.

Examples

			T(3,3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
Triangle starts:
   1;
   1,  2;
   3,  3,  4;
  10, 10,  8,  8;
  36, 36, 29, 20, 16;
		

Links

• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Crossrefs

Cf. A002212, A039919.

Programs

Maple

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z*g/(1-t*z-t*z*g): Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

Formula

T(n,1) = A002212(n-1).

T(n,2) = A002212(n-1) for n >= 3.

Sum_{k=1..n} k*T(n,k) = A039919(n+1).

G.f.: t*z*g/(1 - t*z - t*z*g), where g = 1 + z*g^2 + z*(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).