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A186367
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Number of cycles in all cycle-up-down permutations of {1,2,...,n}. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)<b(2)>b(3)<... .
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2
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1, 3, 10, 38, 165, 812, 4478, 27408, 184529, 1356256, 10809786, 92892928, 856329253, 8430600960, 88292571934, 980197173248, 11499036105537, 142147625652224, 1846872283846922, 25161923756064768, 358706981125488581, 5340498034862030848
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f.: -log(1-sin(z)) / (1-sin(z)).
a(n) = Sum_{k=1..n} k*A186366(n,k).
a(n) ~ n!*n*2^(n+3)/Pi^(n+2)*(2*log(n/Pi) + 2*gamma + 3*log(2) - 2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 02 2013
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EXAMPLE
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a(3) = 10 because the cycle-up-down permutations (1)(2)(3), (12)(3), (13)(2), (1)(23), and (132), have a total of 3+2+2+2+1=10 cycles.
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MAPLE
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g := -ln(1-sin(z))/(1-sin(z)): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 1 .. 22);
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MATHEMATICA
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Rest[CoefficientList[Series[-Log[1-Sin[x]]/(1-Sin[x]), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 02 2013 *)
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PROG
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(PARI) x='x+O('x^30); Vec(serlaplace(-log(1-sin(x))/(1-sin(x)))) \\ G. C. Greubel, Aug 30 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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