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%I #32 Feb 13 2020 17:19:55
%S 1,14,119,798,4655,24794,123970,592020,2731365,12271350,53993940,
%T 233646504,997490844,4211628008,17620076360,73153696336,301758997386,
%U 1237956266316,5054988087457,20558563992050,83322650532485,336691526641470,1356968880100470,5456577564869340,21898107332699325
%N Number of North-East lattice paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly three times.
%C It is related to paired pattern P_3 in Section 3.3 in Pan and Remmel's link.
%H G. C. Greubel, <a href="/A268446/b268446.txt">Table of n, a(n) for n = 6..1000</a>
%H Daniel Birmajer, Juan B. Gil, Michael D. Weiner, <a href="https://arxiv.org/abs/1707.09918">Bounce statistics for rational lattice paths</a>, arXiv:1707.09918 [math.CO], 2017, p. 9.
%H Ran Pan, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%F G.f.: -((2*(2*x + f(x) - 1)^3)/(-2*x + f(x) +1)^4), where f(x) = sqrt(1 - 4*x).
%F From _Karol A. Penson_, Nov 19 2016: (Start)
%F a(n) = 14*binomial(2*n+1, n-6)/(n+8).
%F G.f.: 4^7*z^6/(1+sqrt(1-4*z))^14. - shifted by _Georg Fischer_, Feb 13 2020
%F E.g.f.(in Maple notation): hypergeom([7,15/2],[1,15],4*z).
%F Recurrence: 2*(n+1)*(2*n+3)*a(n)-(n-5)*(n+9)*a(n+1)=0. - _Georg Fischer_, Feb 13 2020
%F Asymptotics: (114688*n-6838272)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2). (End)
%p a:= n-> 14*binomial(2*n+1, n-6)/(n+8): seq(a(n), n=6..23);
%t Rest[Rest[Rest[Rest[Rest[Rest[CoefficientList[Series[-((2 (2 x + Sqrt[1 - 4 x] - 1)^3) / (-2 x + Sqrt[1 - 4 x] + 1)^4), {x, 0, 33}], x]]]]]]] (* _Vincenzo Librandi_, Feb 06 2016 *) (* or *)
%t RecurrenceTable[{2*(n+1)*(2*n+3)*a[n]-(n-5)*(n+9)*a[n+1]==0,a[6]==1},a,{n,6,25}] (* _Georg Fischer_, Feb 13 2020 *)
%o (PARI) for(n=6,25, print1(14*binomial(2*n+1,n-6)/(n+8), ", ")) \\ _G. C. Greubel_, Apr 09 2017, shifted by _Georg Fischer_, Feb 13 2020
%K nonn
%O 6,2
%A _Ran Pan_, Feb 04 2016
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