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)
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
Keywords
Examples
a(3)=3 because we have (1)(2)(3), (132), and (123).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Maple
g := exp(1-cosh(z))/(1-z); gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21); # second Maple program: 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 *) -
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 */
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PARI
my(x='x+O('x^66)); Vec(serlaplace(exp(1-cosh(x))/(1-x))) /* Joerg Arndt, Apr 26 2011 */
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