A268429 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly once.

1, 4, 16, 62, 238, 910, 3475, 13270, 50707, 193948, 742659, 2847126, 10928009, 41993692, 161555008, 622201838, 2398811962, 9257512318, 35760612784, 138263710226, 535038428936, 2072130742074, 8031333322206, 31151602276002, 120915026597458, 469648731423190, 1825348333058230, 7098811400187410, 27623655321103718

Offset: 2

Author

Ran Pan, Feb 04 2016

Keywords

• nonn

Comments

This sequence is related to paired pattern P_2 in Section 3.2 in Pan and Remmel's link.

By symmetry, a(n) is also the number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the left exactly once.

Links

• John Tyler Rascoe, Table of n, a(n) for n = 2..100
• Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Crossrefs

Cf. A000108, A268407, A268429.

Programs

Mathematica

(1 - 4x + x^2 - 2x^3 - Sqrt[1-4x](1 - 2x - 3x^2))/(2(-1 + x(4+x))^2) + O[x]^31 // CoefficientList[#, x]& // Drop[#, 2]& (* Jean-François Alcover, Dec 15 2018 *)

PARI

A_x(N) = {my(x='x+O('x^N), f=sqrt(1-4*x)); Vec((-1+f+2*x)^2/(1-f+(-5+f)*x)^2)}
A_x(31) \\ John Tyler Rascoe, Aug 26 2024

Formula

G.f.: (-1 + f(x) + 2*x)^2/(1 - f(x) + (-5 + f(x))*x)^2, where f(x) = sqrt(1 - 4*x).