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A128724 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k LL's (n >= 0; 0 <= k <= n-2 for n >= 2). 3
1, 1, 3, 9, 1, 30, 5, 1, 107, 23, 6, 1, 399, 105, 31, 7, 1, 1537, 477, 156, 40, 8, 1, 6069, 2166, 771, 219, 50, 9, 1, 24434, 9849, 3772, 1165, 295, 61, 10, 1, 99924, 44869, 18347, 6083, 1676, 385, 73, 11, 1, 413943, 204820, 88908, 31376, 9278, 2322, 490, 86, 12, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 steps in it.

Row 0 and row 1 have one term each; row n has n-1 terms (n >= 2).

Row sums yield A002212.

LINKS

Alois P. Heinz, Rows n = 0..150, flattened

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

FORMULA

T(n,0) = A128725(n).

Sum_{k=0..n-2} k*T(n,k) = A128726(n).

G.f.: G = G(t,z) satisfies z^2*G^3 - z(2 - tz)G^2 + (1 + z - z^2 - tzG + tz - 1 = 0.

EXAMPLE

T(4,1)=5 because we have UDUUUDLL, UUUUDLLD, UUDUUDLL, UUUDUDLL and UUUUDDLL.

Triangle starts:

    1;

    1;

    3;

    9,  1;

   30,  5,  1;

  107, 23,  6,  1;

MAPLE

eq:=z^2*G^3-z*(2-t*z)*G^2+(1+z-z^2-t*z)*G+t*z-1=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 1; for n from 2 to 12 do seq(coeff(P[n], t, j), j=0..n-2) od; # yields sequence in triangular form

# second Maple program:

b:= proc(n, y, t) option remember; expand(`if`(y>n, 0, `if`(n=0, 1,

      `if`(t<0, 0, b(n-1, y+1, 1))+`if`(y<1, 0, b(n-1, y-1, 0))+

      `if`(t>0 or y<1, 0, b(n-1, y-1, -1)*`if`(t<0, z, 1)))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):

seq(T(n), n=0..14);  # Alois P. Heinz, Jun 19 2016

MATHEMATICA

b[n_, y_, t_] := b[n, y, t] = Expand[If[y > n, 0, If[n == 0, 1, If[t < 0, 0, b[n - 1, y + 1, 1]] + If[y < 1, 0, b[n - 1, y - 1, 0]] + If[t > 0 || y < 1, 0, b[n - 1, y - 1, -1]*If[t < 0, z, 1]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Oct 24 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A002212, A128725, A128726.

Sequence in context: A126177 A158483 A128733 * A128753 A179430 A016048

Adjacent sequences:  A128721 A128722 A128723 * A128725 A128726 A128727

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified July 18 19:00 EDT 2019. Contains 325144 sequences. (Running on oeis4.)