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A128733 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k LD's (n>=0; 0<=k<=floor((n-1)/2)). 1
1, 3, 9, 1, 28, 8, 90, 46, 1, 297, 231, 15, 1001, 1079, 138, 1, 3432, 4823, 1006, 24, 11934, 20944, 6388, 320, 1, 41990, 89148, 37026, 3170, 35, 149226, 374034, 201210, 26130, 635, 1, 534888, 1552661, 1042492, 189959, 8170, 48, 1931540, 6393310 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

Row n has ceiling(n/2) terms (n >= 1).

Row sums yield A002212.

LINKS

Table of n, a(n) for n=0..43.

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

FORMULA

T(n,0) = 3(2n)!/((n+2)!(n-1)!) = A000245(n) (n >= 1).

Sum_{k=0..floor((n-1)/2)} k*T(n,k) = A128734(n).

G.f.: G = G(t,z) satisfies tz^2*G^3 - (t-1)z^2*G^2 - (1 - 3z + 2z^2)G + (1-z)^2 = 0.

EXAMPLE

T(3,1)=1 because we have UUUDLD.

Triangle starts:

    1;

    1;

    3;

    9,   1;

   28,   8;

   90,  46,   1;

  297, 231,  15;

MAPLE

eq:=t*z^2*G^3-(t-1)*z^2*G^2-(1-3*z+2*z^2)*G+(1-z)^2=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..floor((n-1)/2)) od; # yields sequence in triangular form

CROSSREFS

Cf. A000245, A002212, A128734.

Sequence in context: A128727 A126177 A158483 * A128724 A128753 A179430

Adjacent sequences:  A128730 A128731 A128732 * A128734 A128735 A128736

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified July 18 07:10 EDT 2019. Contains 325134 sequences. (Running on oeis4.)