A299271 Number of Motzkin paths of length n with all ascents ending at odd heights.

1, 1, 2, 4, 8, 16, 33, 70, 152, 336, 754, 1714, 3940, 9144, 21397, 50428, 119593, 285190, 683422, 1644914, 3974702, 9638478, 23448390, 57213068, 139974092, 343301696, 843911294, 2078912816, 5131312480, 12688618168, 31429741137, 77975828316, 193744122919, 482066579782, 1201045753281, 2996079327262

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..2416

Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.

Formula

G.f.: (1-2*x+2*x^2-2*x^3-sqrt(1-4*x+4*x^2-4*x^4+4*x^5))/(2*(x^2-2*x^3+x^4)).

(6+4*n)*a(n)+(-14-8*n)*a(n+1)+(4*n+8)*a(2+n)+(4*n+24)*a(n+3)+(-50-8*n)*a(n+4)+(35+5*n)*a(n+5)+(-8-n)*a(n+6) = 0. - Robert Israel, Feb 08 2018

Maple

f := gfun:-rectoproc({(6+4*n)*a(n)+(-14-8*n)*a(n+1)+(4*n+8)*a(2+n)+(4*n+24)*a(n+3)+(-50-8*n)*a(n+4)+(35+5*n)*a(n+5)+(-8-n)*a(n+6), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 8, a(5) = 16}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Feb 08 2018

Mathematica

(1 - 2x + 2x^2 - 2x^3 - Sqrt[1 - 4x + 4x^2 - 4x^4 + 4x^5])/(2(x^2 - 2x^3 + x^4)) + O[x]^36 // CoefficientList[#, x]& (* Jean-François Alcover, Sep 14 2018, after Robert Israel *)

Crossrefs

Cf. A299270.

Sequence in context: A129986 A368461 A317880 * A110334 A357904 A084636

Adjacent sequences: A299268 A299269 A299270 * A299272 A299273 A299274

Keyword

nonn

Author

N. J. A. Sloane, Feb 08 2018

Status

approved