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A299270 Number of Motzkin paths of length n with all ascents ending at even heights. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 24 18:55 EST 2021. Contains 341584 sequences. (Running on oeis4.)