0,3

a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 3-cycle.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Christian G. Bower, Table of n, a(n) for n=0..100

a(n) = n! * sum i=0 ... [n/3]( (-1)^i /(i! * 3^i)); a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 3^i) = e^(-1/3); a(n) ~ e^(-1/3) * n!; a(n) ~ e^(-1/3) * (n/e)^n * sqrt(2 * Pi * n). - Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001

a(3) = 4 because the permutations in S_3 that contain no 3-cycles are the trivial permutation and the 3 transpositions.

seq(coeff(convert(series(exp((-x^3)/3)/(1-x), x, 50), polynom), x, i)*i!, i=0..30); # series expansion A000090:=n->n!*add((-1)^i/(i!*3^i), i=0..floor(n/3)); seq(A000090(n), n=0..30); # formula (Pab Ter)

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^3 / 3) + x*O(x^n)) / (1 - x), n))} /* Michael Somos Jul 28 2009 */ - Entry improved by comments from Michael Somos Jul 28 2009

Cf. A000142, A000138, A000266, A060725.

Sequence in context: A025225 A115125 A000831 this_sequence A013115 A007171 A058136

Adjacent sequences: A000087 A000088 A000089 this_sequence A000091 A000092 A000093

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

Entry improved by comments from Michael Somos Jul 28 2009