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

2,2

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

Table of n, a(n) for n=2..30.

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

Formula

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

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 *)

Crossrefs

Cf. A268407.

Sequence in context: A250346 A085781 A113438 * A195339 A172025 A171278

Adjacent sequences: A268426 A268427 A268428 * A268430 A268431 A268432

Keyword

nonn

Author

Ran Pan, Feb 04 2016

Status

approved