1,3

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).

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

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).

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;

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

Cf. A002212, A033321, A128746.

Sequence in context: A264788 A192326 A131733 * A332891 A188524 A126064

Adjacent sequences: A128742 A128743 A128744 * A128746 A128747 A128748

tabl,nonn

Emeric Deutsch, Mar 31 2007

approved