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A268601
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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).
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1
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..30.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
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FORMULA
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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).
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PROG
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(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
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CROSSREFS
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Cf. A268462, A268586, A268587, A268598, A268599.
Sequence in context: A191551 A263627 A172448 * A026577 A204090 A226495
Adjacent sequences: A268598 A268599 A268600 * A268602 A268603 A268604
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KEYWORD
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nonn
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AUTHOR
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Ran Pan, Feb 08 2016
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STATUS
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approved
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