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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
 0, 0, 2, 8, 34, 120, 468, 1680, 6530, 23960, 93532, 348656, 1366260, 5149872, 20238696, 76907808, 302903874, 1158168792, 4569270156, 17555689008, 69356428284, 267518448912, 1058057586456, 4094231982048, 16208177203764, 62887835652720, 249156625186328, 968943740083040, 3841488520364200, 14968574892499040, 59379627044952528 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016. FORMULA a(n) = binomial(2*n,n) - A268600(n). 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). 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 PROG (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 CROSSREFS Cf. A268462, A268586, A268587, A268598, A268599. Sequence in context: A191551 A263627 A172448 * A026577 A204090 A226495 Adjacent sequences:  A268598 A268599 A268600 * A268602 A268603 A268604 KEYWORD nonn AUTHOR Ran Pan, Feb 08 2016 STATUS approved

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Last modified January 22 17:25 EST 2021. Contains 340363 sequences. (Running on oeis4.)