A114627 Number of hill-free Dyck paths of semilength n+3 and having no peaks at level 2 (a Dyck path is said to be hill-free if it has no peaks at level 1).
1, 2, 6, 19, 61, 202, 683, 2348, 8184, 28855, 102731, 368813, 1333684, 4853436, 17761181, 65320691, 241300829, 894958140, 3331323651, 12441078958, 46601721324, 175040968111, 659136721385, 2487852579751, 9410480922018
Offset: 0
Keywords
Examples
a(1)=2 because we have UUUDUDDD and UUUUDDDD, where U=(1,1), D=(1,-1).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Maple
C:=(1-sqrt(1-4*z))/2/z: G:=(C-1)/z/(1+z+z^2-z*(1+z)*C): Gser:=series(G,z=0,32): 1,seq(coeff(Gser,z^n),n=1..28);
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Mathematica
CoefficientList[Series[((1-Sqrt[1-4*x])/2/x-1)/x/(1+x+x^2-x*(1+x)*(1-Sqrt[1-4*x])/2/x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) -
PARI
x='x+O('x^50); Vec((1-2*x-sqrt(1-4*x))/(x^2*(2*x^2+x+1+(1+x)*sqrt(1-4*x)))) \\ G. C. Greubel, Mar 18 2017
Formula
G.f.: (C-1)/(z*(1+z+z^2-z*(1+z)*C)), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) ~ 4^(n+5) / (121*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence +(n+3)*a(n) +(-n+3)*a(n-1) +2*(-5*n-6)*a(n-2) +(-7*n-9)*a(n-3) +2*(-2*n-3)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
Comments