A128719 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UUU's (triplerises) (n >= 0; 0 <= k <= n-2 for n >= 2).

1, 1, 3, 6, 4, 16, 12, 8, 40, 53, 28, 16, 109, 176, 162, 64, 32, 297, 625, 633, 456, 144, 64, 836, 2084, 2677, 2024, 1216, 320, 128, 2377, 7016, 10257, 9849, 6008, 3120, 704, 256, 6869, 23218, 39378, 42222, 32930, 16928, 7776, 1536, 512, 20042, 76811, 146191

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 the path is defined to be the number of its steps.

Row n has n-1 terms (n >= 2).

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) = A128720(n).

Sum_{k=0..n-2} k*T(n,k) = A128721(n) for n >= 2.

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

Example

T(3,1)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
Triangle starts:
   1;
   1;
   3;
   6,  4;
  16, 12,  8;
  40, 53, 28, 16;

Maple

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

Crossrefs

Cf. A002212, A128720, A128721.

Sequence in context: A091808 A357235 A307460 * A145691 A245767 A367302

Adjacent sequences: A128716 A128717 A128718 * A128720 A128721 A128722

Keyword

nonn,tabf

Author

Emeric Deutsch, Mar 30 2007

Status

approved