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%I #14 Sep 08 2022 08:45:56
%S 1,1,1,1,1,2,6,17,44,107,252,588,1376,3245,7717,18485,44535,107796,
%T 261937,638673,1562105,3831655,9423580,23233536,57412612,142174255,
%U 352770105,876922947,2183621209,5446177428,13603846132,34028890577,85234251090,213760737693,536733871490,1349210120198
%N 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).
%C Can be easily expressed using RNA secondary structure terminology.
%H G. C. Greubel, <a href="/A190159/b190159.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A098071(n,0).
%F 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.
%F 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
%F D-finite with recurrence -(n+5)*(n+3)*(n-9)*a(n) +(-n^3-17*n^2-15*n+96)*a(n-1) +3*(7*n^3-43*n^2-20*n-40)*a(n-2) +(34*n^3-214*n^2+357*n-450)*a(n-3) -2*(n-3)*(8*n^2-48*n+183)*a(n-4) +(34*n^3-398*n^2+1461*n-1332)*a(n-5) +3*(7*n^3-83*n^2+220*n+196)*a(n-6) +(-n^3+35*n^2-327*n+822)*a(n-7) -(n-11)*(n+3)*(n-9)*a(n-8)=0. - _R. J. Mathar_, Jul 24 2022
%e 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.
%p 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);
%t 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 *)
%o (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
%o (Magma) m:=40; R<x>:=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
%Y Cf. A098071, A004148
%K nonn
%O 0,6
%A _Emeric Deutsch_, May 05 2011
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