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A268600 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
1, 2, 4, 12, 36, 132, 456, 1752, 6340, 24660, 91224, 356776, 1337896, 5250728, 19877904, 78209712, 298176516, 1175437428, 4505865144, 17789574792, 68490100536, 270739425528, 1046041377264, 4139198745552, 16039426479336, 63522770785032, 246761907761776, 977995685565072, 3807202080396240, 15098691607042000, 58884954519908896 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 even. Details can be found in Section 4.1 in Pan and Remmel's link.

LINKS

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.

FORMULA

a(n) = binomial(2*n,n) - A268601(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).

MATHEMATICA

CoefficientList[ Series[1/(2 Sqrt[1 - 4x]) + 1/(4 - 2 Sqrt[1 + 4x]), {x, 0, 25}], x] (* Robert G. Wilson v, Nov 24 2016 *)

PROG

(PARI) x = 'x + O('x^40); 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: A009152 A081443 A086647 * A187119 A062161 A195733

Adjacent sequences:  A268597 A268598 A268599 * A268601 A268602 A268603

KEYWORD

nonn

AUTHOR

Ran Pan, Feb 08 2016

STATUS

approved

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Last modified September 26 06:24 EDT 2017. Contains 292502 sequences.