

A128743


Number of UU's (i.e., doublerises) in all skew Dyck paths of semilength n.


2



0, 0, 2, 13, 69, 346, 1700, 8286, 40264, 195488, 949302, 4613025, 22436997, 109240038, 532410060, 2597468685, 12684628125, 62002335160, 303332650190, 1485213237135, 7277719953415, 35687662907750, 175120787451540
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OFFSET

0,3


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203.


FORMULA

a(n) = Sum_{k=0..n1} k*A128718(n,k).
G.f.: (14*z+z^2+(z1)*sqrt(16*z+5*z^2))/(2*z*sqrt(16*z+5*z^2)).
a(n) ~ 3*5^(n1/2)/(2*sqrt(Pi*n)).  Vaclav Kotesovec, Mar 20 2014
Conjecture: (n+1)*(n2)^2*a(n) (n1)*(6*n^215*n+4)*a(n1) +5*(n2)*(n1)^2*a(n2)=0.  R. J. Mathar, Jun 17 2016
Conjecture verified using the differential equation 4*g(z)+(20*z^3+2*z^22*z)*g'(z)+(25*z^415*z^3)*g''(z)+(5*z^56*z^4+z^3)*g'''(z)=0 satisfied by the G.f.  Robert Israel, Dec 25 2017


EXAMPLE

a(2)=2 because the paths of semilength 2 are UDUD, UUDD and UUDL, having altogether 2 UU's.


MAPLE

G:=(14*z+z^2+(z1)*sqrt(16*z+5*z^2))/2/z/sqrt(16*z+5*z^2): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..25);


MATHEMATICA

CoefficientList[Series[(14*x+x^2+(x1)*Sqrt[16*x+5*x^2])/2/x/Sqrt[16*x+5*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)


PROG

(PARI) z='z+O('z^50); concat([0, 0], Vec((14*z+z^2+(z1)*sqrt(16*z+5*z^2))/(2*z*sqrt(16*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017


CROSSREFS

Cf. A128718.
Sequence in context: A097977 A136780 A301944 * A218184 A264735 A289926
Adjacent sequences: A128740 A128741 A128742 * A128744 A128745 A128746


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Mar 30 2007


STATUS

approved



