A184958 - OEIS

%I #29 Apr 13 2017 10:54:02

%S 0,0,0,0,5,25,269,1883,20103,180927,2172149,23893639,326640467,

%T 4246326071,65675585793,985133786895,17069814958319,290186854291423,

%U 5579050805341613,106001965301490647,2241684406438644939,47075372535211543719

%N 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.

%H Alois P. Heinz, <a href="/A184958/b184958.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = Sum_{k>=0} k*A186769(n,k).

%F E.g.f.: (1/2) * (2*(1-cosh(z)) - log(1-z^2))/(1-z).

%F 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

%F 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

%e a(4) = 5 because the only permutations of {1,2,3,4} having nonincreasing even cycles are (1243), (1324), (1342), (1423), and (1432).

%p 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);

%t 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 *)

%t 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 *)

%Y Cf. A186761, A186763, A186764, A186766, A186768, A186769.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Feb 27 2011