A268402 - OEIS

%I #8 Oct 07 2016 10:22:39

%S 1,7,40,204,977,4493,20091,88025,379766,1618898,6835636,28640302,

%T 119236085,493772409,2035611612,8359873866,34219553297,139672169795,

%U 568675783762,2310315996126,9367885987455,37920179012135,153263612914150,618611076034828,2493830719572639,10042451847789161

%N Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly four times.

%C This sequence is related to paired pattern P_2 in Pan and Remmel's link.

%C By symmetry, it is also the number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the left exactly four times.

%H Ran Pan and 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.: -((-1 + f(x))^5*x^3*(-1 + f(x) + 2*x))/(2*(1 - f(x) + (-5 + f(x))*x)^5), where f(x) = sqrt(1 - 4*x).

%F Conjecture: -(n-3)*(n-5)*(55*n-618)*a(n) +(-55*n^3-840*n^2+6969*n-5742)*a(n-1)

%F   +(3135*n^3 -37532*n^2 +138815*n -163614)*a(n-2) +(-7645*n^3 +93072*n^2 -343985*n +391386)*a(n-3) -18*(n-3)*(55*n-126)*(2*n-5)*a(n-4)=0. - _R. J. Mathar_, Oct 07 2016

%Y Cf. A268400, A268401.

%K nonn

%O 5,2

%A _Ran Pan_, Feb 03 2016