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 A190159 Number of peakless Motzkin paths of length n and having no uhh...hd's starting at level 0, where u = (1, 1), h = (1, 0) and d = (1, -1). 2
 1, 1, 1, 1, 1, 2, 6, 17, 44, 107, 252, 588, 1376, 3245, 7717, 18485, 44535, 107796, 261937, 638673, 1562105, 3831655, 9423580, 23233536, 57412612, 142174255, 352770105, 876922947, 2183621209, 5446177428, 13603846132, 34028890577, 85234251090, 213760737693, 536733871490, 1349210120198 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Can be easily expressed using RNA secondary structure terminology. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA a(n) = A098071(n,0). G.f.=G=G(z) satisfies aG^2 + bG + c = 0, where a=z^2*(1-z-z^2+2z^3), b=-(1-z)(1-2z+2z^2+z^3), c=(1-z)^2. G.f.: ((1 - x)/(2*x^2*(1 - x - x^2 + 2*x^3)))*(1 - 2*x + 2*x^2 + x^3 - sqrt((1 - 2*x + 2*x^2 + x^3)^2 - 4*x^2*(1 - x - x^2 + 2*x^3))). - G. C. Greubel, Oct 22 2018 EXAMPLE a(5)=2 because among the 8 (=A004148(5)) peakless Motzkin paths of length 5 only hhhhh and uuhdd have no subword of type uhh...hd starting at level 0. MAPLE eq := z^2*(-z^2+2*z^3-z+1)*G^2-(1-z)*(z^3+2*z^2-2*z+1)*G+(1-z)^2 = 0: g := RootOf(eq, G): Gser := series(g, z=0, 40): seq(coeff(Gser, z, n), n=0..35); MATHEMATICA With[{a = x^2*(1 - x - x^2 + 2*x^3)}, CoefficientList[Series[((1 - x)/(2*a))*(1 - 2*x + 2*x^2 + x^3 - Sqrt[(1 - 2*x + 2*x^2 + x^3)^2 - 4*a]), {x, 0, 40}], x]] (* G. C. Greubel, Oct 22 2018 *) PROG (PARI) x='x+O('x^40); Vec(((1-x)/(2*x^2*(1-x-x^2+2*x^3)))*(1-2*x+2*x^2 + x^3 - sqrt((1-2*x+2*x^2+x^3)^2 -4*x^2*(1-x-x^2+2*x^3)))) \\ G. C. Greubel, Oct 22 2018 (MAGMA) m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(((1 -x)/(2*x^2*(1 -x -x^2 +2*x^3)))*(1 -2*x +2*x^2 +x^3 -Sqrt((1-2*x+2*x^2 + x^3)^2 -4*x^2*(1 -x -x^2 +2*x^3))))); // G. C. Greubel, Oct 22 2018 CROSSREFS Cf. A098071, A004148 Sequence in context: A014833 A019439 A018024 * A000996 A309757 A020963 Adjacent sequences:  A190156 A190157 A190158 * A190160 A190161 A190162 KEYWORD nonn AUTHOR Emeric Deutsch, May 05 2011 STATUS approved

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Last modified June 17 19:09 EDT 2021. Contains 345085 sequences. (Running on oeis4.)