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A128735 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k LDU's (n >= 0; 0 <= k <= floor((n-1)/3) for n >= 1). 2
1, 3, 10, 35, 1, 127, 10, 474, 69, 1810, 406, 3, 7043, 2193, 49, 27839, 11252, 496, 111503, 55858, 3996, 12, 451640, 271206, 28120, 270, 1847032, 1296584, 181027, 3575, 7616692, 6130552, 1094856, 36300, 55, 31637664, 28753124, 6325592, 312832 (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 the path is defined to be the number of its steps.

Row n has ceiling(n/3) terms (n >= 1).

Row sums yield A002212.

T(n,0) = A128736(n).

Apparently, T(3k+1,k) = binomial(3k,k)/(2k+1) = A001764(k).

LINKS

Table of n, a(n) for n=0..38.

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

FORMULA

Sum_{k>=0} k*T(n,k) = A128737(n).

G.f.: G = G(t,z) satisfies (t+1)zG^3 - (2 - 4z + 3tz)G^2 + 3(1 - 2z + tz)G - 1 + 2z - tz = 0.

EXAMPLE

T(7,2)=3 because we have UUUD(LDU)UUD(LDU)D, UUUUD(LDU)UD(LDU)D and UUUUUD(LDU)D(LDU)D (the LDU's are shown between parentheses).

Triangle starts:

     1;

     1;

     3;

    10;

    35,   1;

   127,  10;

   474,  69;

  1810, 406,   3;

MAPLE

eq:=(t+1)*z*G^3-(2-4*z+3*t*z)*G^2+3*(1-2*z+t*z)*G-1+2*z-t*z=0: G:=RootOf(eq, G): Gser:=simplify(series(G, z=0, 18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: 1; for n from 1 to 15 do seq(coeff(P[n], t, j), j=0..floor((n-1)/3)) od; # yields sequence in triangular form

CROSSREFS

Cf. A001764, A002212, A128736, A128737.

Sequence in context: A149034 A149035 A099907 * A330050 A159309 A112107

Adjacent sequences:  A128732 A128733 A128734 * A128736 A128737 A128738

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Mar 31 2007

STATUS

approved

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Last modified December 5 12:24 EST 2021. Contains 349557 sequences. (Running on oeis4.)