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%I #12 Jun 07 2016 13:52:40
%S 1,18,189,1518,10350,63180,356265,1893294,9612108,47071640,223926516,
%T 1040310648,4739192952,21238169904,93865125915,409972529754,
%U 1772528290407,7596549816030,32308782859535,136496564854650,573285572389530,2395339717603140,9962435643667605
%N Number of North-East lattice paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly four times.
%C It is related to paired pattern P_3 in Section 3.3 in Pan and Remmel's link.
%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*(-1 + f(x) + 2*x)^4)/(1 + f(x) - 2*x)^5, where f(x) = sqrt(1 - 4*x).
%F Conjecture: -(n+10)*(n-8)*a(n) +2*n*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jun 07 2016
%t Rest[Rest[Rest[Rest[Rest[Rest[Rest[Rest[CoefficientList[Series[(2 (-1 + Sqrt[1 - 4 x] + 2 x)^4) / (1 + Sqrt[1 - 4 x] - 2 x)^5, {x, 0, 33}], x]]]]]]]]] (* _Vincenzo Librandi_, Feb 06 2016 *)
%o (PARI) x='x+O('x^100); Vec((2*(-1 + (1 - 4*x)^(1/2) + 2*x)^4)/(1 + (1 - 4*x)^(1/2) - 2*x)^5) \\ _Altug Alkan_, Feb 04 2016
%Y Cf. A268446.
%K nonn
%O 8,2
%A _Ran Pan_, Feb 04 2016
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