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 A257300 Number of Motzkin paths of length n with no peaks at level 2. 3
 1, 1, 2, 4, 8, 17, 38, 88, 210, 514, 1285, 3270, 8447, 22100, 58455, 156077, 420153, 1139155, 3108095, 8527675, 23514124, 65127571, 181111940, 505487115, 1415502195, 3975790024, 11197966459, 31619946886, 89496047586, 253858251337, 721531869889, 2054639741185 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA G.f.: 1/(1-x-x^2/(1-x+x^2*(1-M(x)))), where M(x) is the g.f. of Motzkin numbers A001006. a(n) ~ 3^(n+7/2) / (50*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015 Conjecture: n*a(n) +(-5*n+3)*a(n-1) +6*(n)*a(n-2) +2*(n-9)*a(n-3) +6*(-n+4)*a(n-4) +(n-6)*a(n-5) +3*(-n+3)*a(n-6)=0. - R. J. Mathar, Sep 24 2016 EXAMPLE For n=4 we have 8 paths: HHHH, UDUD, UHDH, HUHD, UHHD, UDHH, HUDH and HHUD. MATHEMATICA CoefficientList[Series[1/(1-x-x^2/(1-x+x^2*(1-(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *) PROG (PARI) x='x + O('x^50); Vec(1/(1-x-x^2/(1-x+x^2*(1-(1-x-sqrt(1-2*x-3*x^2))/(2*x^2))))) \\ G. C. Greubel, Feb 14 2017 CROSSREFS Cf. A089372. Sequence in context: A119685 A025276 A006461 * A229202 A003007 A086615 Adjacent sequences:  A257297 A257298 A257299 * A257301 A257302 A257303 KEYWORD nonn AUTHOR José Luis Ramírez Ramírez, Apr 20 2015 STATUS approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)