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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).
1
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
OFFSET
1,3
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 sums yield A002212.
T(n,1) = A033321(n-1).
Sum_{k=1..n} k*T(n,k) = A128746(n).
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
FORMULA
G.f.: t*z/(1 - z*g - 2*t*z), where g = 1 + z*g^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
KEYWORD
tabl,nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved