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A128737
Number of LDU's in all skew Dyck paths of semilength n.
1
0, 0, 0, 0, 1, 10, 69, 412, 2291, 12244, 63886, 328256, 1669363, 8429384, 42349096, 211982828, 1058244079, 5272285552, 26227527576, 130323237088, 647013004499, 3210128312122, 15919166804461, 78915323039268, 391100149306301
OFFSET
0,6
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.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = Sum_{k>=0} k*A128735(n,k).
G.f.: z(g-1)^3/(4g - 2zg - 6zg^2 - 3 + 3*z), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
Conjecture D-finite with recurrence: -10*(n+1)*(n-4)*a(n) +(73*n^2-273*n+140)*a(n-1) +(-132*n^2+641*n-734) *a(n-2) +(n-3)*(89*n-269)*a(n-3) -20*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 17 2016
EXAMPLE
a(4)=1 because among the 36 (=A002212(4)) skew Dyck paths of semilength 4 only UUUDLDUD has a LDU.
MAPLE
g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: ser:=series(z*(g-1)^3/(4*g-2*z*g-6*z*g^2-3+3*z), z=0, 30): seq(coeff(ser, z, n), n=0..27);
CROSSREFS
Cf. A128735.
Sequence in context: A081280 A038806 A016273 * A320228 A130548 A160662
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved