

A128743


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


1



0, 0, 2, 13, 69, 346, 1700, 8286, 40264, 195488, 949302, 4613025, 22436997, 109240038, 532410060, 2597468685, 12684628125, 62002335160, 303332650190, 1485213237135, 7277719953415, 35687662907750, 175120787451540
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

a(n)=Sum[k*A128718(n,k), k=0..n1].


LINKS

Table of n, a(n) for n=0..22.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

G.f.=[14z+z^2+(z1)sqrt(16z+5z^2)]/[2z*sqrt(16z+5z^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


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 *)


CROSSREFS

Cf. A128718.
Sequence in context: A038144 A097977 A136780 * A218184 A264735 A188676
Adjacent sequences: A128740 A128741 A128742 * A128744 A128745 A128746


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Mar 30 2007


STATUS

approved



