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A268601 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). 1

%I

%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).

%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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Last modified December 16 19:57 EST 2018. Contains 318188 sequences. (Running on oeis4.)