A299270 Number of Motzkin paths of length n with all ascents ending at even heights.

1, 1, 1, 1, 2, 5, 12, 27, 60, 135, 309, 716, 1673, 3935, 9311, 22154, 52977, 127255, 306913, 742918, 1804301, 4395371, 10737206, 26296601, 64555741, 158825720, 391551973, 967118177, 2392964346, 5930752193, 14721605128, 36595817145, 91096419441, 227054764556, 566615061751, 1415614697677, 3540584874294, 8864485647609

Offset

0,5

Links

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

Yan Zhuang, A generalized Goulden-Jackson cluster method and lattice path enumeration, arXiv:1508.02793 [math.CO], 2015.

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

Formula

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

(6+4*n)*a(n) + (-18-8*n)*a(n+1) + (18+8*n)*a(2+n) + 12*a(n+3) + (-54-8*n)*a(n+4) + (9*n+63)*a(n+5) + (-39-5*n)*a(n+6) + (9+n)*a(n+7) = 0. - Robert Israel, Feb 09 2018

Maple

f := gfun:-rectoproc({(6+4*n)*a(n)+(-18-8*n)*a(n+1)+(18+8*n)*a(2+n)+12*a(n+3)+(-54-8*n)*a(n+4)+(9*n+63)*a(n+5)+(-39-5*n)*a(n+6)+(9+n)*a(n+7), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 5, a(6) = 12}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Feb 09 2018

Mathematica

CoefficientList[Series[(1 - 2  x + 2  x^2 - Sqrt[1 - 4  x + 4  x^2 - 4  x^4 + 4  x^5]) / (2  (x^2 - x^3 + x^4)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 09 2018 *)

Crossrefs

Cf. A299271.

Sequence in context: A000102 A304175 A086589 * A190171 A091596 A077863

Adjacent sequences: A299267 A299268 A299269 * A299271 A299272 A299273

Keyword

nonn

Author

N. J. A. Sloane, Feb 08 2018

Status

approved