A128751 Number of ascents of length at least 2 in all skew Dyck paths of semilength n.

1, 1, 1, 2, 1, 9, 1, 29, 6, 1, 83, 53, 1, 226, 294, 22, 1, 602, 1319, 297, 1, 1588, 5244, 2362, 90, 1, 4171, 19302, 14464, 1649, 1, 10935, 67379, 75505, 17155, 394, 1, 28645, 226321, 353721, 133395, 9153, 1, 75012, 738324, 1532222, 862950, 117903, 1806, 1

Offset

0,4

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 a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps.

Row sums yield A002212.

Links

Table of n, a(n) for n=0..49.

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

Formula

T(n,0) = 1.

Sum_{k>=0} k*T(n,k) = A128752(n).

G.f.: G = G(t,z) satisfies z(1 - z + tz)G^2 - (1 - z + z^2 - tz^2)G + 1 - z = 0.

Example

T(4,2)=6 because we have (UU)DD(UU)DD, (UU)DD(UU)DL, (UU)D(UU)LLL, (UU)D(UU)DLD, (UU)D(UU)DDL and (UU)D(UU)DLL (the ascents of length at least 2 are shown between parentheses).
Triangle starts:
  1;
  1;
  1,   2;
  1,   9;
  1,  29,   6;
  1,  83,  53;
  1, 226, 294,  22;

Maple

eq:=z*(1-z+t*z)*G^2-(1-z+z^2-t*z^2)*G+1-z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 15 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form

Crossrefs

Cf. A002212, A128752.

Sequence in context: A261124 A100945 A133399 * A129168 A360733 A353204

Adjacent sequences: A128748 A128749 A128750 * A128752 A128753 A128754

Keyword

nonn,tabf

Author

Emeric Deutsch, Mar 31 2007

Status

approved