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, 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

Offset: 0

Author

Ran Pan, Feb 08 2016

Keywords

• nonn

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.

Examples

			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
		

Links

• Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Crossrefs

Cf. A268462, A268586, A268587, A268598, A268599.

Cf. A000984, A126984.

Programs

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 *)

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

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).

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

a(n) = (-1)^n*A126984(n) + A268601(n). - Michael Somos, May 16 2022