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%I #25 May 16 2022 17:38:34
%S 1,2,4,12,36,132,456,1752,6340,24660,91224,356776,1337896,5250728,
%T 19877904,78209712,298176516,1175437428,4505865144,17789574792,
%U 68490100536,270739425528,1046041377264,4139198745552,16039426479336,63522770785032,246761907761776,977995685565072,3807202080396240,15098691607042000,58884954519908896
%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 even. 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) - A268601(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
%F a(n) = (-1)^n*A126984(n) + A268601(n). - _Michael Somos_, May 16 2022
%e G.f. = 1 + 2*x + 4*x^2 + 12*x^3 + 36*x^4 + 132*x^5 + 456*x^6 + ... - _Michael Somos_, May 16 2022
%t CoefficientList[ Series[1/(2 Sqrt[1 - 4x]) + 1/(4 - 2 Sqrt[1 + 4x]), {x, 0, 25}], x] (* _Robert G. Wilson v_, Nov 24 2016 *)
%o (PARI) my(x = 'x + O('x^40)); 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.
%Y Cf. A000984, A126984.
%K nonn
%O 0,2
%A _Ran Pan_, Feb 08 2016
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