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A186765
Number of permutations of {1,2,...,n} having no increasing even cycles. 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.
6
1, 1, 1, 3, 14, 70, 419, 2933, 23421, 210789, 2108144, 23189584, 278279165, 3617629145, 50646737049, 759701055735, 12155215581362, 206638664883154, 3719496008830391, 70670424167777429, 1413408484443295197, 29681578173309199137, 652994719769134284068
OFFSET
0,4
COMMENTS
a(n) = A186764(n,0).
LINKS
FORMULA
E.g.f.: exp(1-cosh(z))/(1-z).
a(n)=((sum(m=1..n,sum(k=1..m,((-1)^k*sum(j=0..k,((sum(i=0..j,(j-2*i)^m*binomial(j, i)))*(-1)^(j-k)*binomial(k, j))/2^j))/k!)/m!))+1)*n! [From Vladimir Kruchinin, Apr 25 2011]
a(n) ~ n! * exp(1-cosh(1)). - Vaclav Kotesovec, Feb 24 2014
EXAMPLE
a(3)=3 because we have (1)(2)(3), (132), and (123).
MAPLE
g := exp(1-cosh(z))/(1-z); gser := series(g, z = 0, 27); seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21)
MATHEMATICA
CoefficientList[Series[E^(1-Cosh[x])/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 24 2014 *)
PROG
(Maxima)
a(n):=((sum(sum(((-1)^k*sum(((sum((j-2*i)^m*binomial(j, i), i, 0, j))*(-1)^(j-k)*binomial(k, j))/2^j, j, 0, k))/k!, k, 1, m)/m!, m, 1, n))+1)*n!; [Vladimir Kruchinin, Apr 25 2011]
(PARI) x='x+O('x^66); Vec(serlaplace(exp(1-cosh(x))/(1-x))) /* Joerg Arndt, Apr 26 2011 */
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 27 2011
STATUS
approved