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.

0, 0, 0, 0, 5, 25, 269, 1883, 20103, 180927, 2172149, 23893639, 326640467, 4246326071, 65675585793, 985133786895, 17069814958319, 290186854291423, 5579050805341613, 106001965301490647, 2241684406438644939, 47075372535211543719

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