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%I A268399 #8 Jun 07 2016 13:52:47
%S A268399 14,70,286,1099,4124,15327,56770,210188,779076,2893111,10767680,
%T A268399 40171225,150229560,563151435,2115877410,7967261640,30063189300,
%U A268399 113663662560,430549220244,1633782030774,6210024076424,23641792007350,90140083306676,344168324083080,1315850249846440,5037257160310193
%N A268399 Number of North-East lattice paths from (0,0) to (n,n) that have exactly four east steps below the subdiagonal y = x-1.
%C A268399 This sequence is related to paired pattern P_1 in Pan and Remmel's link.
%H A268399 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 A268399 G.f.: -((-1 + f(x) + 2*x)^2*(-1 + f(x) + x*(2*f(x) + x + 5*f(x)*x - 10*x^2)))/(8*x^2), where f(x) = sqrt(1 - 4*x).
%F A268399 Conjecture: -(n+2)*(641*n-5292)*a(n) +(-641*n^2-1681*n+22692)*a(n-1) +(-4067*n^2+10502*n+80184) *a(n-2) +6*(5629*n-11668)*(2*n-11) *a(n-3)=0. - _R. J. Mathar_, Jun 07 2016
%F A268399 Conjecture: +(n+2)*(1023*n^2-11311*n+23730) *a(n) +(1023*n^3+15038*n^2-85751*n+44490)*a(n-1) -10*(2*n-9) *(1023*n^2-2563*n+1450)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2016
%Y A268399 Cf. A268370.
%K A268399 nonn
%O A268399 5,1
%A A268399 _Ran Pan_, Feb 03 2016

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