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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..49.

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

Cf. A002212, A128752.

Sequence in context: A261124 A100945 A133399 * A129168 A293416 A194555

Adjacent sequences:  A128748 A128749 A128750 * A128752 A128753 A128754

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified June 1 12:52 EDT 2020. Contains 334762 sequences. (Running on oeis4.)