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 A184958 Number of nonincreasing even cycles in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)
 0, 0, 0, 0, 5, 25, 269, 1883, 20103, 180927, 2172149, 23893639, 326640467, 4246326071, 65675585793, 985133786895, 17069814958319, 290186854291423, 5579050805341613, 106001965301490647, 2241684406438644939, 47075372535211543719 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..450 FORMULA a(n) = Sum_{k>=0} k*A186769(n,k). E.g.f.: (1/2) * (2*(1-cosh(z)) - log(1-z^2))/(1-z). a(n) ~ n!/2 * (log(n/2) - 1/exp(1) + 2 - exp(1) + gamma), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 05 2013 a(n) = (n!*H(n)-2F0(1,-n;;-1) + (-1)^(n+1)*2F0(1,-n;;1)+n!*(2+(-1)^n*LerchPhi(-1,1,n+1)-log(2)))/2, where H(n) is the n-th harmonic number. - Benedict W. J. Irwin, May 30 2016 EXAMPLE a(4) = 5 because the only permutations of {1,2,3,4} having nonincreasing even cycles are (1243), (1324), (1342), (1423), and (1432). MAPLE g := (1/2*(2*(1-cosh(z))-ln(1-z^2)))/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21); MATHEMATICA With[{nn=30}, CoefficientList[Series[1/2(2(1-Cosh[x])-Log[1-x^2])/(1-x), {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Oct 22 2011 *) Table[(n! HarmonicNumber[n] - HypergeometricPFQ[{1, -n}, {}, -1] + (-1)^(n + 1) HypergeometricPFQ[{1, -n}, {}, 1] + n! (2 + (-1)^n LerchPhi[-1, 1, 1 + n] - Log))/2, {n, 0, 20}] (* Benedict W. J. Irwin, May 30 2016 *) CROSSREFS Cf. A186761, A186763, A186764, A186766, A186768, A186769. Sequence in context: A061839 A143600 A209529 * A145076 A185063 A165656 Adjacent sequences:  A184955 A184956 A184957 * A184959 A184960 A184961 KEYWORD nonn AUTHOR Emeric Deutsch, Feb 27 2011 STATUS approved

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Last modified November 20 05:07 EST 2019. Contains 329323 sequences. (Running on oeis4.)