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A128750
Number of skew Dyck paths of semilength n having no ascents of length 1.
4
1, 0, 2, 4, 14, 44, 150, 520, 1850, 6696, 24602, 91500, 343846, 1303572, 4979822, 19150352, 74075890, 288022160, 1125076210, 4413061972, 17375007294, 68641377980, 272014578822, 1081009104664, 4307221752874, 17203123381304
OFFSET
0,3
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.
Hankel transform is 2^ceiling(n(n+1)/3). Binomial transform is A059278. - Paul Barry, Feb 11 2009
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
FORMULA
a(n) = A128749(n,0).
G.f.: G = G(z) satisfies z(1 + z)G^2 - (1 - z^2)G + 1 - z = 0.
G.f.: 1/(1+x-x/(1-x-x/(1+x-x/(1-x-x/(1+x-x/(1-... (continued fraction). - Paul Barry, Feb 11 2009
From Paul Barry, Feb 11 2009: (Start)
G.f.: (1/(1+x))c(x/(1-x^2)) where c(x) is the g.f. of A000108;
G.f.: 1/(1-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-.... (continued fraction);
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(floor((n+k)/2),k)*A000108(k).
(End)
Conjecture: (n+1)*a(n) +(-4*n+3)*a(n-1) +(-2*n-1)*a(n-2) +(4*n-11)*a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2012
a(n) ~ sqrt(5+3*sqrt(5)) * (2+sqrt(5))^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 08 2014
EXAMPLE
a(3)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
MAPLE
G:=(1-z^2-sqrt((1-z^2)*(1-4*z-z^2)))/2/z/(1+z): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..30);
MATHEMATICA
CoefficientList[Series[(1-x^2-Sqrt[(1-x^2)*(1-4*x-x^2)])/2/x/(1+x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
CROSSREFS
Cf. A128749.
Sequence in context: A169982 A367101 A243323 * A047152 A007866 A226909
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved