login
A128747
Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k peaks of height >1 (n >= 1; 0 <= k <= n-1).
1
1, 1, 2, 1, 7, 2, 1, 18, 15, 2, 1, 41, 68, 25, 2, 1, 88, 244, 171, 37, 2, 1, 183, 765, 866, 351, 51, 2, 1, 374, 2199, 3651, 2355, 636, 67, 2, 1, 757, 5954, 13601, 12708, 5421, 1058, 85, 2, 1, 1524, 15438, 46355, 58977, 36198, 11116, 1653, 105, 2, 1, 3059, 38747, 147768
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.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
FORMULA
T(n,0) = 1.
Sum_{k=0..n-1} k*T(n,k) = A128748(n).
G.f.: G(t,z) = (1 - z + z*K(t,z))/(1 - z*K(t,z)) - 1, where K = K(t,z) satisfies zK^2 - (1 - tz)K + 1 - z = 0 (K is the g.f. for the number of peaks; see A126182).
EXAMPLE
T(3,1)=7 because we have UDU(UD)D, UDU(UD)L, U(UD)DUD, UU(UD)DD, UU(UD)LD, UU(UD)DL and UU(UD)LL (the peaks of height >1 are shown between parentheses).
Triangle starts:
1;
1, 2;
1, 7, 2;
1, 18, 15, 2;
1, 41, 68, 25, 2;
MAPLE
K:=(1-z*t-sqrt(z^2*t^2-2*z*t+1+4*z^2-4*z))/2/z: G:=z*(2*K-1)/(1-z*K): Gser:=simplify(series(G, z=0, 14)): 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=0..n-1) od; # yields sequence in triangular form
CROSSREFS
Sequence in context: A369924 A347800 A341738 * A205945 A124392 A144446
KEYWORD
tabl,nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved