

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 <= n1).


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
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OFFSET

1,3


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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) 21912203.


FORMULA

T(n,0) = 1.
Sum_{k=0..n1} 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:=(1z*tsqrt(z^2*t^22*z*t+1+4*z^24*z))/2/z: G:=z*(2*K1)/(1z*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..n1) 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



