A129159 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having abscissa of the first return to the x-axis equal to 2k (1 <= k <= n).

1, 2, 1, 4, 4, 2, 11, 9, 11, 5, 37, 21, 31, 34, 14, 138, 59, 76, 116, 112, 42, 544, 198, 198, 315, 448, 384, 132, 2220, 743, 599, 825, 1358, 1758, 1353, 429, 9286, 2964, 2091, 2345, 3724, 5922, 6963, 4862, 1430, 39588, 12251, 8026, 7604, 10388, 17304, 25872

Offset

1,2

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) = 1 + A002212(n-1) (indeed, the path U^nDL^(n-1) and the paths UDP, where P is a skew Dyck path of semilength n-1).

T(n,n) = binomial(2n-2,n-1)/n = A000108(n-1) (the Catalan numbers).

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

Sum_{k=1..n} k*T(n,k) = A129160(n).

G.f.: tzhg + z(h-1), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2)) and h = 1 + tzh^2 + z(h-1) (h = h(t,z) is the g.f. for skew Dyck paths according to the semi-abscissa of the last point on the x-axis and semilength; see A108198).

Example

T(3,2)=4 because we have UUDDUD, UUUDLD, UUDUDL and UUUDDL.
Triangle starts:
   1;
   2,  1;
   4,  4,  2;
  11,  9, 11,  5;
  37, 21, 31, 34, 14;

Maple

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: h:=(1-z-sqrt(z^2-2*z+1+4*t*z^2-4*t*z))/2/t/z: G:=t*z*h*g+z*(h-1): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 11 do P[n]:=sort(expand(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. A000108, A002212, A108198, A129160.

Sequence in context: A051289 A134308 A090802 * A095830 A193915 A101621

Adjacent sequences: A129156 A129157 A129158 * A129160 A129161 A129162

Keyword

nonn,tabl

Author

Emeric Deutsch, Apr 03 2007

Status

approved