%I A000090 M1295 N0496
%S A000090 1,1,2,4,16,80,520,3640,29120,259840,2598400,28582400,343235200,
%T A000090 4462057600,62468806400,936987251200,14991796019200,254860532326400,
%U A000090 4587501779660800,87162533813555200,1743250676271104000
%N A000090 E.g.f. exp((-x^3)/3)/(1-x).
%C A000090 a(n) is the number of permutations in the symmetric group S_n whose cycle 
               decomposition contains no 3-cycle.
%D A000090 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000090 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000090 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               85.
%D A000090 R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 
               93, problem 7.
%H A000090 Christian G. Bower, <a href="b000090.txt">Table of n, a(n) for n=0..100</
               a>
%F A000090 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
%e A000090 a(3) = 4 because the permutations in S_3 that contain no 3-cycles are 
               the trivial permutation and the 3 transpositions.
%p A000090 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)
%o A000090 (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
%Y A000090 Cf. A000142, A000138, A000266, A060725.
%Y A000090 Sequence in context: A025225 A115125 A000831 this_sequence A013115 A007171 
               A058136
%Y A000090 Adjacent sequences: A000087 A000088 A000089 this_sequence A000091 A000092 
               A000093
%K A000090 nonn,easy
%O A000090 0,3
%A A000090 N. J. A. Sloane (njas(AT)research.att.com).
%E A000090 More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
%E A000090 Entry improved by comments from Michael Somos Jul 28 2009