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

Author

Emeric Deutsch, Mar 31 2007

Keywords

• nonn
• tabf

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.

Examples

			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;
		

Links

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

Crossrefs

Cf. A002212, A128752.

Programs

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

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.