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 A299271 Number of Motzkin paths of length n with all ascents ending at odd heights. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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: A004149 A129986 A317880 * A110334 A084636 A161869 Adjacent sequences:  A299268 A299269 A299270 * A299272 A299273 A299274 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 08 2018 STATUS approved

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Last modified January 17 10:59 EST 2019. Contains 319218 sequences. (Running on oeis4.)