|
Search: id:A128745
|
|
|
| A128745 |
|
Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having height of the last peak equal to k (1<=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. |
|
+0
2
|
|
| 1, 1, 2, 2, 4, 4, 6, 10, 12, 8, 21, 32, 36, 32, 16, 79, 116, 124, 112, 80, 32, 311, 448, 468, 416, 320, 192, 64, 1265, 1800, 1860, 1640, 1280, 864, 448, 128, 5275, 7440, 7640, 6720, 5280, 3712, 2240, 1024, 256, 22431, 31426, 32136, 28256, 22336, 16032, 10304
(list; table; graph; listen)
|
|
|
OFFSET
|
1,3
|
|
|
COMMENT
|
Row sums yield A002212. T(n,1)=A033321(n-1). Sum(k*T(n,k),k=1..n)=A128746(n).
|
|
REFERENCES
|
E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).
|
|
FORMULA
|
G.f.= tz/(1-zg-2tz), where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)]/(2z).
|
|
EXAMPLE
|
T(3,2)=4 because we have UDUUDD, UDUUDL, UUDUDD and UUDUDL.
Triangle starts:
1;
1,2;
2,4,4;
6,10,12,8;
21,32,36,32,16;
|
|
MAPLE
|
g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=t*z/(1-2*t*z-z*g): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
|
|
CROSSREFS
|
Cf. A002212, A033321, A128746.
Sequence in context: A045674 A143483 A131733 this_sequence A126064 A066813 A033732
Adjacent sequences: A128742 A128743 A128744 this_sequence A128746 A128747 A128748
|
|
KEYWORD
|
tabl,nonn
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
|
|
|
Search completed in 0.002 seconds
|
