OFFSET
0,2
COMMENTS
Number of degree-2n permutations without odd cycles and with odd number of even cycles, offset 1. E.g.f.: x^2/(2*sqrt(1-x^2)). - Vladeta Jovovic, Aug 10 2007
REFERENCES
Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, p. 219.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..224
FORMULA
a(n) ~ (2*n)^(2*n+2)/exp(2*n). - Vaclav Kotesovec, Oct 24 2013
EXAMPLE
E.g.f. = 1/2!*x^2 + 6/4!*x^4 + 135/6!*x^6 + 6300/8!*x^8 + ...
MAPLE
b:= proc(n, t) option remember; `if`(n=0, t, add(
b(n-2*j, 1-t)*binomial(n-1, 2*j-1)*(2*j-1)!, j=1..n/2))
end:
a:= n-> b(2*n+2, 0):
seq(a(n), n=0..14); # Alois P. Heinz, Mar 30 2026
MATHEMATICA
nn = 30; Select[Range[0, nn]! CoefficientList[Series[Sinh[Log[1/(1 - x^2)^(1/2)]], {x, 0, nn}], x], # > 0 &] (* Geoffrey Critzer, Jan 15 2012 *)
With[{nn=30}, Take[-CoefficientList[Series[Sinh[Log[x+1]-ArcTanh[x]], {x, 0, nn}], x] Range[0, nn]!, {3, -1, 2}]] (* Harvey P. Dale, Oct 30 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved
