A128745 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the last peak equal to k (1 <= k <= n).

1, 1, 2, 2, 4, 4, 6, 10, 12, 8, 21, 32, 36, 32, 16, 79, 116, 124, 112, 80, 32, 311, 448, 468, 416, 320, 192, 64, 1265, 1800, 1860, 1640, 1280, 864, 448, 128, 5275, 7440, 7640, 6720, 5280, 3712, 2240, 1024, 256, 22431, 31426, 32136, 28256, 22336, 16032, 10304

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.

T(n,1) = A033321(n-1).

Sum_{k=1..n} k*T(n,k) = A128746(n).

Links

Table of n, a(n) for n=1..52.

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

Formula

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

Example

T(3,2)=4 because we have UDUUDD, UDUUDL, UUDUDD and UUDUDL.
Triangle starts:
   1;
   1,  2;
   2,  4,  4;
   6, 10, 12,  8;
  21, 32, 36, 32, 16;

Maple

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z/(1-2*t*z-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, A033321, A128746.

Sequence in context: A264788 A192326 A131733 * A332891 A188524 A126064

Adjacent sequences: A128742 A128743 A128744 * A128746 A128747 A128748

Keyword

tabl,nonn

Author

Emeric Deutsch, Mar 31 2007

Status

approved