A000138 Expansion of e.g.f. exp(-x^4/4)/(1-x).
1, 1, 2, 6, 18, 90, 540, 3780, 31500, 283500, 2835000, 31185000, 372972600, 4848643800, 67881013200, 1018215198000, 16294848570000, 277012425690000, 4986223662420000, 94738249585980000, 1894745192712372000, 39789649046959812000, 875372279033115864000
Offset: 0
Examples
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.
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
- Simon Plouffe, Exact formulas for integer sequences
Programs
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Mathematica
nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x^4/4]/(1-x),{x,0,nn}],x] (* Geoffrey Critzer, Oct 28 2012 *) -
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 */
Formula
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
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
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
Extensions
Entry improved by comments from Michael Somos, Jul 28 2009
Name corrected by Joerg Arndt, May 27 2011
Comments