OFFSET
1,3
REFERENCES
Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 189, Exercise 19 for k=1. With (-1)^k omitted.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..440
FORMULA
a(n) = (n!/2)*Sum_{j=0..floor(n/2)-1} (-1)^j/(j!*2^j), n >= 1.
E.g.f.: x^2/(1-x)/2*exp(-x^2/2). - Vladeta Jovovic, Nov 09 2003
From Paul Weisenhorn, Jun 02 2010: (Start)
In general, for k cycles of length 2,
a(n) = n!*Sum_{j=k..floor(n/2)} (-1)^j/((j-k)!*2^j*k!).
G.f.: (exp(-z^2/2)*z^2*k)/((1-z)*2^k*k!). (End)
a(n) ~ exp(-1/2)/2 * n!. - Vaclav Kotesovec, Mar 18 2014
EXAMPLE
From Bernard Schott, Feb 19 2019: (Start)
For S_4, the six permutations that have exactly one transposition in their cycle decomposition are (12)(3)(4), (13)(2)(4), (14)(2)(3), (23)(1)(4), (24)(1)(3), (34)(1)(2).
For S_5, there are exactly 10 transpositions: (12), (13), (14), (15), (23), (24), (25), (34), (35), (45), and for each transposition, there are 3 permutations that have exactly this transposition and no other transposition in their cycle decomposition; for example, for transposition (12), these three permutations: (12)(3)(4)(5), (12)(345), (12)(354), so a(5) = 10 * 3 = 30. (End)
MAPLE
G=(exp(-z^2/2)*z^2*k)/((1-z)*2^k*k!): Gser=series(G, z=0, 21):
for n from 2*k to 20 do a(n)=n!*coeff(Gser, z, n): end do: # Paul Weisenhorn, Jun 02 2010
MATHEMATICA
d=Exp[-x^2/2]/(1-x); Range[0, 20]! CoefficientList[Series[(x^2/2! )d, {x, 0, 20}], x] (* Geoffrey Critzer, Nov 29 2011 *)
PROG
(PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace( x^2*exp(-x^2/2)/(2*(1-x)) ))) \\ G. C. Greubel, Feb 19 2019
(Magma) m:=32; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x^2*Exp(-x^2/2)/(2*(1-x)) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Feb 19 2019
(Sage) m = 30; T = taylor(x^2*exp(-x^2/2)/(2*(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (1..m)] # G. C. Greubel, Feb 19 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003
EXTENSIONS
More terms from Wolfdieter Lang, Feb 22 2008
STATUS
approved