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%I #12 Jan 30 2020 21:29:17
%S 0,0,2,8,34,120,468,1680,6530,23960,93532,348656,1366260,5149872,
%T 20238696,76907808,302903874,1158168792,4569270156,17555689008,
%U 69356428284,267518448912,1058057586456,4094231982048,16208177203764,62887835652720,249156625186328,968943740083040,3841488520364200,14968574892499040,59379627044952528
%N Expansion of 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).
%C a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which the total number of east steps below y = x - 1 or above y = x + 1 is odd. Details can be found in Section 4.1 in Pan and Remmel's link.
%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 a(n) = binomial(2*n,n) - A268600(n).
%F G.f.: 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).
%F Conjecture D-finite with recurrence: 3*n*(n-1)*a(n) -8*(n-1)*(5*n-12)*a(n-1) +4*(28*n-73)*a(n-2) +160*(2*n-5)*(2*n-7)*a(n-3) -192*(2*n-5)*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020
%o (PARI) x = 'x + O('x^30); concat(vector(2), Vec(1/(2*sqrt(1-4*x)) - 1/(4 - 2*sqrt(1+4*x)))) \\ _Michel Marcus_, Feb 11 2016
%Y Cf. A268462, A268586, A268587, A268598, A268599.
%K nonn
%O 0,3
%A _Ran Pan_, Feb 08 2016
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