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 A257290 Number of 3-Motzkin paths of length n with no level steps at even level. 3
 1, 0, 1, 3, 11, 39, 140, 504, 1823, 6621, 24144, 88380, 324699, 1197045, 4427565, 16427385, 61129025, 228103185, 853399640, 3200710680, 12032399045, 45332769075, 171148151095, 647412581643, 2453529142471, 9314461044639, 35419207688050, 134894888442714, 514506926871927 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{i=0..floor(n/2)} 3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108. G.f.: (1 - 3*z - sqrt((1-3*z)*(1-3*z-4*z^2)))/(2*z^2). a(n) ~ sqrt(5) * 4^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015 Conjecture: (n+2)*a(n) +3*(-2*n-1)*a(n-1) +5*(n-1)*a(n-2) +6*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 24 2016 EXAMPLE For n=3 we have 3 paths: UH1D, UH2D, UH3D. MATHEMATICA CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *) PROG (PARI) x='x+O('x^50); Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Feb 14 2017 CROSSREFS Cf. A090345, A025266. Sequence in context: A289834 A007482 A134760 * A281482 A132889 A245390 Adjacent sequences:  A257287 A257288 A257289 * A257291 A257292 A257293 KEYWORD nonn AUTHOR José Luis Ramírez Ramírez, Apr 20 2015 STATUS approved

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Last modified July 17 18:47 EDT 2019. Contains 325109 sequences. (Running on oeis4.)