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A128746
Height of the last peak summed over all skew Dyck paths of semilength n.
2
1, 5, 22, 94, 401, 1723, 7475, 32749, 144803, 645627, 2900256, 13115820, 59669295, 272918415, 1254314310, 5789850730, 26831078075, 124785337255, 582247766810, 2724905891890, 12787603121195, 60162698218325, 283715348775727
OFFSET
1,2
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.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = Sum_{k=1,..,n} A128745(n,k).
G.f.: 2*z*(1+z+sqrt(1-6*z+5*z^2))/(1-3*z+sqrt(1-6*z+5*z^2))^2.
a(n) ~ 5^(n+3/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: -(n+2)*(n-1)*a(n) +(6*n^2-3*n+2)*a(n-1) -5*n*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 08 2015
EXAMPLE
a(2)=5 because the skew Dyck paths of semilength 2 are UD(UD), U(UD)D and U(UD)L and their last peaks (shown between parentheses) have heights 1, 2 and 2, respectively.
MAPLE
G:=2*z*(1+z+sqrt(1-6*z+5*z^2))/(1-3*z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..27);
MATHEMATICA
Rest[CoefficientList[Series[2*x*(1+x+Sqrt[1-6*x+5*x^2])/(1-3*x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) z='z+O('z^50); Vec(2*z*(1+z+sqrt(1-6*z+5*z^2))/(1-3*z + sqrt(1-6*z+5*z^2))^2) \\ G. C. Greubel, Mar 20 2017
CROSSREFS
Cf. A128745.
Sequence in context: A026672 A049652 A026877 * A049675 A053154 A141222
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved