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 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). 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Column 0 of A114626. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 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 EXAMPLE a(1)=2 because we have UUUDUDDD and UUUUDDDD, where U=(1,1), D=(1,-1). 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); 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 *) PROG (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 CROSSREFS Cf. A114626. Sequence in context: A228180 A035929 A071646 * A289591 A148464 A148465 Adjacent sequences: A114624 A114625 A114626 * A114628 A114629 A114630 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 18 2005 STATUS approved

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Last modified March 31 05:21 EDT 2023. Contains 361634 sequences. (Running on oeis4.)