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A166294 Number of peaks at even 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, 4, 12, 34, 92, 242, 628, 1616, 4138, 10570, 26970, 68798, 175545, 448176, 1145058, 2927924, 7493021, 19191836, 49195806, 126205062, 324000494, 832371414, 2139802870, 5504256592, 14166936063, 36483006046, 94000206216 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = Sum_{k=0..n-1} k*A166293(n,k).

G.f.: G=z[z - 1 + (1 - z + z^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

EXAMPLE

a(3)=4 because the paths UDUDUD, UDU(UD)D, U(UD)DUD, and U(UD)(UD)D have 0 + 1 + 1 + 2 = 4 peaks at even 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*(z-1+(1-z+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 = 1 .. 30);

MATHEMATICA

Rest[CoefficientList[Series[x*(x-1+(1-x+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, A166292, A166293.

Sequence in context: A107069 A191823 A110335 * A307305 A176753 A248873

Adjacent sequences:  A166291 A166292 A166293 * A166295 A166296 A166297

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 12 2009

STATUS

approved

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Last modified February 17 12:14 EST 2020. Contains 331996 sequences. (Running on oeis4.)