A128722 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k hills (i.e., peaks at level 1) (0 <= k <= n).

1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 22, 9, 4, 0, 1, 84, 35, 12, 5, 0, 1, 334, 138, 49, 15, 6, 0, 1, 1368, 563, 198, 64, 18, 7, 0, 1, 5734, 2352, 825, 264, 80, 21, 8, 0, 1, 24480, 10015, 3504, 1121, 336, 97, 24, 9, 0, 1, 106086, 43308, 15123, 4833, 1452, 414, 115, 27, 10, 0, 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.

T(n,0) = A128723(n).

Row sums yield A002212.

Sum_{k=0..n} k*T(n,k) = A033321(n).

Links

Table of n, a(n) for n=0..65.

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

Formula

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

Example

T(3,1)=3 because we have (UD)UUDD, (UD)UUDL and UUDD(UD) (the hills are shown between parentheses).
Triangle starts:
   1;
   0,  1;
   2,  0,  1;
   6,  3,  0,  1;
  22,  9,  4,  0,  1;
  84, 35, 12,  5,  0,  1;

Maple

g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-z+z*g)/(1+z-z*g-t*z): Gser:=simplify(series(G, z=0, 14)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form

Crossrefs

Cf. A002212, A033321, A128723.

Sequence in context: A365770 A059299 A332673 * A324659 A304485 A289537

Adjacent sequences: A128719 A128720 A128721 * A128723 A128724 A128725

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 30 2007

Status

approved