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

1,3

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.

Links

Table of n, a(n) for n=1..53.

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

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).

Example

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;

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

Crossrefs

Cf. A002212, A039919.

Sequence in context: A111574 A330510 A173590 * A293984 A207608 A290818

Adjacent sequences: A128741 A128742 A128743 * A128745 A128746 A128747

Keyword

tabl,nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved