A129168 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k UU's starting at level 0 (i.e., doublerises at level 1; n >= 0, 0 <= k <= floor(n/2)).

1, 1, 1, 2, 1, 9, 1, 33, 2, 1, 119, 17, 1, 443, 97, 2, 1, 1716, 477, 25, 1, 6884, 2205, 193, 2, 1, 28403, 9947, 1203, 33, 1, 119811, 44539, 6695, 321, 2, 1, 514370, 199465, 34934, 2425, 41, 1, 2240032, 896375, 175494, 15833, 481, 2, 1, 9870894, 4047160, 861739

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

Row n contains 1 + floor(n/2) terms.

Row sums yield A002212.

Links

Table of n, a(n) for n=0..52.

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

Formula

Sum_{k=0..floor(n/2)} k*T(n,k) = A129169(n).

G.f.: G(t,z) = (2 + t - 3tz - t*sqrt(1 - 6z + 5z^2))/(2 - t - 2z + 3tz + t*sqrt(1 - 6z + 5z^2)).

Example

T(4,2)=2 because we have UUDDUUDD and UUDDUUDL.
Triangle starts:
  1;
  1;
  1,   2;
  1,   9;
  1,  33,   2;
  1, 119,  17;
  1, 443,  97,   2;

Maple

G:=(2+t-3*t*z-t*sqrt(1-6*z+5*z^2))/(2-t-2*z+3*t*z+t*sqrt(1-6*z+5*z^2)): Gser:=simplify(series(G, z=0, 17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form

Crossrefs

Cf. A002212, A129169.

Sequence in context: A100945 A133399 A128751 * A360733 A353204 A293416

Adjacent sequences: A129165 A129166 A129167 * A129169 A129170 A129171

Keyword

nonn,tabf

Author

Emeric Deutsch, Apr 05 2007

Status

approved