

A128749


Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 1.


1



1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 14, 12, 9, 0, 1, 44, 53, 25, 14, 0, 1, 150, 196, 132, 44, 20, 0, 1, 520, 777, 555, 269, 70, 27, 0, 1, 1850, 3064, 2486, 1260, 485, 104, 35, 0, 1, 6696, 12233, 10902, 6264, 2496, 804, 147, 44, 0, 1, 24602, 49096, 47955, 30108, 13600
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OFFSET

0,4


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 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..59.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203.


FORMULA

T(n,0) = A128750(n).
Sum_{k=0..n} k*T(n,k) = A085362(n1).
G.f.: G = G(t,z) satisfies z(1 + z  tz)G^2  (1  tz + tz^2  z^2)G + 1  z = 0.


EXAMPLE

T(3,1)=5 because we have (U)DUUDD, (U)DUUDL, UUDD(U)D, UUD(U)DD and UUD(U)DL (the ascents of length 1 are shown between parentheses).
Triangle starts:
1;
0, 1;
2, 0, 1;
4, 5, 0, 1;
14, 12, 9, 0, 1;
44, 53, 25, 14, 0, 1;


MAPLE

eq:=z*(1+zt*z)*G^2(1t*z+t*z^2z^2)*G+1z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form


CROSSREFS

Cf. A002212, A085362, A128750.
Sequence in context: A077909 A247126 A229223 * A106579 A287318 A173003
Adjacent sequences: A128746 A128747 A128748 * A128750 A128751 A128752


KEYWORD

tabl,nonn


AUTHOR

Emeric Deutsch, Mar 31 2007


STATUS

approved



