0,4

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.

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

T(n,0) = A128750(n).

Sum_{k=0..n} k*T(n,k) = A085362(n-1).

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

T(3,1)=5 because we have (U)DUUDD, (U)DUUDL, UUDD(U)D, UUD(U)DD and UUD(U)DL (the ascents of length 1 are shown between parentheses).

Triangle starts:

1;

0, 1;

2, 0, 1;

4, 5, 0, 1;

14, 12, 9, 0, 1;

44, 53, 25, 14, 0, 1;

eq:=z*(1+z-t*z)*G^2-(1-t*z+t*z^2-z^2)*G+1-z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 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

Cf. A002212, A085362, A128750.

Sequence in context: A327117 A359107 A229223 * A106579 A378318 A287318

Adjacent sequences: A128746 A128747 A128748 * A128750 A128751 A128752

tabl,nonn

Emeric Deutsch, Mar 31 2007

approved