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A128751
Number of ascents of length at least 2 in all skew Dyck paths of semilength n.
1
1, 1, 1, 2, 1, 9, 1, 29, 6, 1, 83, 53, 1, 226, 294, 22, 1, 602, 1319, 297, 1, 1588, 5244, 2362, 90, 1, 4171, 19302, 14464, 1649, 1, 10935, 67379, 75505, 17155, 394, 1, 28645, 226321, 353721, 133395, 9153, 1, 75012, 738324, 1532222, 862950, 117903, 1806, 1
OFFSET
0,4
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 a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U 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} k*T(n,k) = A128752(n).
G.f.: G = G(t,z) satisfies z(1 - z + tz)G^2 - (1 - z + z^2 - tz^2)G + 1 - z = 0.
EXAMPLE
T(4,2)=6 because we have (UU)DD(UU)DD, (UU)DD(UU)DL, (UU)D(UU)LLL, (UU)D(UU)DLD, (UU)D(UU)DDL and (UU)D(UU)DLL (the ascents of length at least 2 are shown between parentheses).
Triangle starts:
1;
1;
1, 2;
1, 9;
1, 29, 6;
1, 83, 53;
1, 226, 294, 22;
MAPLE
eq:=z*(1-z+t*z)*G^2-(1-z+z^2-t*z^2)*G+1-z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 15 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form
CROSSREFS
Sequence in context: A261124 A100945 A133399 * A129168 A360733 A353204
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved