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A166292 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). 4
0, 1, 2, 5, 12, 29, 72, 181, 460, 1178, 3030, 7815, 20188, 52193, 134992, 349205, 903398, 2337135, 6046310, 15642402, 40469824, 104708914, 270937964, 701129755, 1814581514, 4696886211, 12159165336, 31481922733, 81523933604, 211143257951 (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

a(n) = Sum_{k=0,..,n} k*A166291(n,k).

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.

a(n) ~ sqrt(55 + 123/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+5/2)). - Vaclav Kotesovec, Mar 20 2014

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

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

EXAMPLE

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).

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A004148, A166291, A166293, A166294

Sequence in context: A227237 A005593 A122745 * A010374 A307788 A025273

Adjacent sequences:  A166289 A166290 A166291 * A166293 A166294 A166295

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 12 2009

STATUS

approved

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Last modified October 2 05:52 EDT 2022. Contains 357191 sequences. (Running on oeis4.)