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A000266 E.g.f. exp(-x^2/2) / (1-x).
(Formerly M2991 N1211)
15
1, 1, 1, 3, 15, 75, 435, 3045, 24465, 220185, 2200905, 24209955, 290529855, 3776888115, 52876298475, 793144477125, 12690313661025, 215735332237425, 3883235945814225, 73781482970470275, 1475629660064134575, 30988222861346826075, 681740902935880863075 (list; graph; refs; listen; history; text; internal format)
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

T. D. Noe, Table of n, a(n) for n=0..100

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 ... [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

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

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

See also A000138 and A000090.

Cf. A130905, A193385.

Sequence in context: A005053 A183411 A136778 * A059838 A079164 A240941

Adjacent sequences:  A000263 A000264 A000265 * A000267 A000268 A000269

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

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

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Last modified October 21 17:46 EDT 2014. Contains 248377 sequences.