A128746 Height of the last peak summed over all skew Dyck paths of semilength n.
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
Keywords
Examples
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.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
Crossrefs
Cf. A128745.
Programs
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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);
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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 *) -
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
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
Comments