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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 (list; table; 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..59.

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

FORMULA

T(n,0) = A128750(n).

Sum_{k=0..n} k*T(n,k) = A085362(n-1).

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+z-t*z)*G^2-(1-t*z+t*z^2-z^2)*G+1-z=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

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Last modified June 17 22:17 EDT 2019. Contains 324200 sequences. (Running on oeis4.)