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%I #11 Jul 26 2022 15:05:54
%S 0,1,2,5,12,29,72,181,460,1178,3030,7815,20188,52193,134992,349205,
%T 903398,2337135,6046310,15642402,40469824,104708914,270937964,
%U 701129755,1814581514,4696886211,12159165336,31481922733,81523933604,211143257951
%N Number of peaks at odd level in all Dyck paths of semilength n with no UUU's and no DDD's, (U=(1,1), D=(1,-1)). These Dyck paths are counted by the secondary structure numbers (A004148).
%H G. C. Greubel, <a href="/A166292/b166292.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0,..,n} k*A166291(n,k).
%F G.f.: G=[z^2 + 3z - 2 + (z^4 + z^3 - z^2 - 4z + 2)g(z)]/[(1 - z - z^2)(1 - z - z^2 - 2z^3*g(z)], where g=g(z) satisfies g = 1 + zg + z^2*g + z^3*g^2.
%F a(n) ~ sqrt(55 + 123/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+5/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F Equivalently, a(n) ~ phi^(2*n + 5) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021
%F D-finite with recurrence -(n+3)*(12263959*n-453745718)*a(n) +(-12263959*n^2-1898630196*n-3269930920)*a(n-1) +2*(201299201*n^2-33603137*n-693679841)*a(n-2) +4*(-226348358*n^2+2057096353*n-415271997)*a(n-3) +(282398701*n^2-4452810911*n+4125507350)*a(n-4) +(46943591*n^2-3487699726*n+8812781352)*a(n-5) +4*(214523727*n^2-2147935054*n+5837738425)*a(n-6) +2*(-101430217*n^2+3147700949*n-14145800266)*a(n-7) -5*(76453537*n-336152052)*(n-7)*a(n-8) +(n-8)*(126836791*n-1033250150)*a(n-9)=0. - _R. J. Mathar_, Jul 26 2022
%e a(3)=5 because the paths (UD)(UD)(UD), (UD)UUDD, UUDD(UD), and UUDUDD have 3 + 1 + 1 + 0 peaks at odd level (shown between parentheses).
%p g := ((1-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^3: G := (z^2+3*z-2+(z^4+z^3-z^2-4*z+2)*g)/((1-z-z^2)*(1-z-z^2-2*z^3*g)): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 30);
%t CoefficientList[Series[(x^2+3*x-2+(x^4+x^3-x^2-4*x+2)*((1-x-x^2-Sqrt[1-2*x-x^2-2*x^3+x^4])*1/2)/x^3)/((1-x-x^2)*(1-x-x^2-2*x^3*((1-x-x^2-Sqrt[1-2*x-x^2-2*x^3+x^4])*1/2)/x^3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%Y Cf. A004148, A166291, A166293, A166294
%K nonn
%O 0,3
%A _Emeric Deutsch_, Oct 12 2009
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