

A128753


Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDUDU's (n >= 0; 0 <= k <= n2 for n >= 2).


0



1, 1, 3, 9, 1, 31, 4, 1, 113, 19, 4, 1, 431, 86, 21, 4, 1, 1697, 393, 101, 23, 4, 1, 6847, 1800, 492, 116, 25, 4, 1, 28161, 8279, 2388, 596, 131, 27, 4, 1, 117631, 38218, 11603, 3032, 705, 146, 29, 4, 1, 497665, 177013, 56407, 15403, 3732, 819, 161, 31, 4, 1
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OFFSET

0,3


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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.
Rows 0 and 1 have one term each; row n has n1 terms (n >= 2).
Row sums yield A002212.


LINKS

Table of n, a(n) for n=0..56.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203.


FORMULA

T(n,0) = A052709(n+1).
Sum_{k=0..n2} k*T(n,k) = A026376(n2).
G.f.: G = G(t,z) satisfies z(1 + z  tz)G^2  (1  tz)G + 1  tz = 0. G = C((1+ztz)/(1tz)), where C(z) = (1  sqrt(1  4z))/(2z) is the Catalan function.


EXAMPLE

T(4,1)=4 because we have (UDUDU)UDD, (UDUDU)UDL, U(UDUDU)DD and U(UDUDU)DL (the subwords UDUDU are shown between parentheses).
Triangle starts
1;
1;
3;
9, 1;
31, 4, 1;
113, 19, 4, 1;


MAPLE

C:=z>(1sqrt(14*z))/2/z: G:=C(z*(1+zt*z)/(1t*z)): 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..n2) od; # yields sequence in triangular form


CROSSREFS

Cf. A002212, A052709, A026376.
Sequence in context: A158483 A128733 A128724 * A179430 A016048 A256501
Adjacent sequences: A128750 A128751 A128752 * A128754 A128755 A128756


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Apr 01 2007


STATUS

approved



