A186768 Number of nonincreasing odd 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 odd if it has an odd number of entries.

0, 0, 0, 1, 4, 43, 258, 2525, 20200, 222119, 2221190, 28061889, 336742668, 4856656283, 67993187962, 1107076110629, 17713217770064, 322047491979087, 5796854855623566, 116542615962575753, 2330852319251515060, 51380800712458456259

Offset

0,5

Comments

a(n) = Sum(k*A186766(n,k), k>=0).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

E.g.f.: g(z)=[log((1+z)/(1-z))-2sinh(z)]/(2(1-z)).

a(n) ~ n!/2 * (log(2*n) + gamma - exp(1) + exp(-1)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 07 2013

Example

a(3)=1 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 0+0+0+0+0+1 =1 increasing odd cycles.

Maple

g := ((ln((1+z)/(1-z))-2*sinh(z))*1/2)/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);

Mathematica

CoefficientList[Series[(Log[(1+x)/(1-x)]-2*Sinh[x])/(2*(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)

Crossrefs

Cf. A186761, A186763, A186764, A186766, A186769, A184958.

Sequence in context: A296683 A297649 A186678 * A130545 A027311 A198205

Adjacent sequences: A186765 A186766 A186767 * A186769 A186770 A186771

Keyword

nonn

Author

Emeric Deutsch, Feb 27 2011

Extensions

Typo in e.g.f. corrected by Vaclav Kotesovec, Oct 07 2013

Status

approved