|
| |
|
|
A129154
|
|
Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive Dyck factors (n>=0; 0<=k<=n). 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. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.
|
|
3
| |
|
|
1, 0, 1, 1, 1, 1, 4, 3, 2, 1, 15, 11, 6, 3, 1, 59, 41, 22, 10, 4, 1, 241, 159, 84, 38, 15, 5, 1, 1011, 639, 331, 150, 60, 21, 6, 1, 4326, 2640, 1342, 606, 246, 89, 28, 7, 1, 18797, 11146, 5570, 2500, 1023, 380, 126, 36, 8, 1, 82685, 47884, 23567, 10503, 4312, 1630
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,7
|
|
|
COMMENTS
| Row sums yield A002212. T(n,0)=A129155(n). Sum(k*T(n,k),k=0..n)=A129156(n).
|
|
|
LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2192-2203
|
|
|
FORMULA
| G.f.=G(t,z)=[1+z(g-1)]/[1-z(g-C)-tzC], where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)]/(2z) and C=1+zC^2=[1-sqrt(1-4z)]/(2z) is the Catalan function.
|
|
|
EXAMPLE
| T(4,2)=6 because we have (UD)(UD)UUDL, (UD)(UUDUDD), (UD)(UUUDDD), (UUDD)(UUDD), (UUDUDD)(UD) and (UUUDDD)(UD) (the primitive Dyck factors are shown between parentheses).
Triangle starts:
1;
0,1;
1,1,1;
4,3,2,1;
15,11,6,3,1;
|
|
|
MAPLE
| G:=(3-3*z-sqrt(1-6*z+5*z^2))/(2-t+z+(t-1)*sqrt(1-4*z)+sqrt(1-6*z+5*z^2)): Gser:=simplify(series(G, z=0, 14)): 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
|
|
|
CROSSREFS
| Cf. A002212, A129155, A129156.
Sequence in context: A194743 A113778 A155172 * A055115 A108438 A082504
Adjacent sequences: A129151 A129152 A129153 * A129155 A129156 A129157
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2007
|
| |
|
|
