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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)<b(2)<b(3)<... .  A cycle is said to be even if it has an even number of entries. 4
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]))/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.)