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%I M1635 N0638 #49 Mar 04 2024 19:36:38
%S 1,1,2,6,18,90,540,3780,31500,283500,2835000,31185000,372972600,
%T 4848643800,67881013200,1018215198000,16294848570000,277012425690000,
%U 4986223662420000,94738249585980000,1894745192712372000,39789649046959812000,875372279033115864000
%N Expansion of e.g.f. exp(-x^4/4)/(1-x).
%C a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 4-cycle.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.
%H T. D. Noe, <a href="/A000138/b000138.txt">Table of n, a(n) for n = 0..100</a>
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/exact.htm">Exact formulas for integer sequences</a>
%F a(n) = n! * Sum_{i=0..floor(n/4)} (-1)^i / (i! * 4^i); a(n)/n! ~ Sum_{i >= 0} (-1)^i / (i! * 4^i) = e^(-1/4); a(n) ~ e^(-1/4) * n!; a(n) ~ e^(-1/4) * (n/e)^n * sqrt(2*Pi*n). - Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
%F a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), here k=4, n>=0. _Simon Plouffe_, from old notes, 1993
%F E.g.f.: exp(-x^4/4)/(1-x) = 1/G(0); G(k) = 1 - x/(1 - (x^3)/(x^3 - 4*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Feb 28 2012
%e a(4) = 18 because in S_4 the permutations with no 4-cycle are the complement of the six 4-cycles so a(4) = 4! - 6 = 18.
%t nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x^4/4]/(1-x),{x,0,nn}],x] (* _Geoffrey Critzer_, Oct 28 2012 *)
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^4/4) + x*O(x^n)) / (1 - x), n))} /* _Michael Somos_, Jul 28 2009 */
%Y Cf. A000142, A000090, A000266, A060725, A060726, A060727.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E Entry improved by comments from _Michael Somos_, Jul 28 2009
%E Name corrected by _Joerg Arndt_, May 27 2011
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