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

Table of n, a(n) for n=1..59.

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

Cf. A002212, A128748.

Sequence in context: A280691 A092666 A019426 * A205945 A124392 A144446

Adjacent sequences:  A128744 A128745 A128746 * A128748 A128749 A128750

KEYWORD

tabl,nonn

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified November 22 00:32 EST 2019. Contains 329383 sequences. (Running on oeis4.)