OFFSET
0,4
COMMENTS
a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no transposition.
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
Larry Carter and Stan Wagon, The Mensa Correctional Institute, The American Mathematical Monthly 125.4 (2018): 306-319.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 104
Simon Plouffe, Exact formulas for integer sequences, March 1993.
FORMULA
E.g.f.: exp( x + Sum_{k>2} x^k / k ). - Michael Somos, Jul 25 2011
a(n) = n! * Sum_{i=0..floor(n/2)} (-1)^i /(i! * 2^i); a(n)/n! ~ Sum_{i>=0} (-1)^i /(i! * 2^i) = e^(-1/2); a(n) ~ e^(-1/2) * n!; a(n) ~ e^(-1/2) * (n/e)^n * sqrt(2*Pi*n). - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
A027616(n) + a(n) = n!. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003
a(n) = n!*floor((floor(n/2)! * 2^floor(n/2) / exp(1/2) + 1/2)) / (floor(n/2)! * 2^floor(n/2)), n >= 0. - Simon Plouffe from old notes, 1993
E.g.f.: 1/(1-x)*exp(-(x^2)/2) = 1/((1-x)*G(0)); G(k) = 1+(x^2)/(2*(2*k+1)-2*(x^2)*(2*k+1)/((x^2)+4*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
E.g.f.: 1/Q(0), where Q(k) = 1 - x/(1 - x/(x - (2*k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
D-finite with recurrence: a(n) - n*a(n-1) + (n-1)*a(n-2) - (n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Feb 16 2020
EXAMPLE
a(3) = 3 because the permutations in S_3 that contain no transpositions are the trivial permutation and the two 3-cycles.
MAPLE
G:=exp(-z^2/2)/(1-z): Gser:=series(G, z=0, 26): for n from 0 to 25 do a(n):=n!*coeff(Gser, z, n): end do: seq(a(n), n=0..20); # Paul Weisenhorn, May 29 2010
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-j)*(j-1)!*binomial(n-1, j-1), j=[1, $3..n]))
end:
seq(a(n), n=0..30); # Alois P. Heinz, May 12 2016
MATHEMATICA
a=Log[1/(1-x)]-x^2/2; Range[0, 20]! CoefficientList[Series[Exp[a], {x, 0, 20}], x] (* Geoffrey Critzer, Nov 29 2011 *)
PROG
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp(-(x^2/2)+x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Christian G. Bower
Entry improved by comments from Michael Somos, Jul 28 2009
Minor editing by Johannes W. Meijer, Jul 25 2011
STATUS
approved