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.

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

Links

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

Formula

a(n) = A186764(n,0).

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!. [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);
# Alternative:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*((j-1)!+irem(j, 2)-1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Feb 05 2025

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) my(x='x+O('x^66)); Vec(serlaplace(exp(1-cosh(x))/(1-x))) /* Joerg Arndt, Apr 26 2011 */

Crossrefs

Column k=0 of A186764.

Cf. A186761, A186762, A186766, A186767, A186769, A186770

Sequence in context: A006772 A320421 A009020 * A109792 A137177 A158196

Adjacent sequences: A186762 A186763 A186764 * A186766 A186767 A186768

Keyword

nonn

Author

Emeric Deutsch, Feb 27 2011

Status

approved