OFFSET
0,4
COMMENTS
The number of simple labeled graphs on n nodes whose connected components are stars. - Geoffrey Critzer, Dec 10 2011
Equivalently, the number of minimal edge covers of the complete graph K_n. - Andrew Howroyd, Aug 04 2017
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(b).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Minimal Edge Cover
FORMULA
a(n) = n!*Sum_{k=1..n} (1/k!)*( binomial(k, n-k)*2^(k-n)*(-1)^k + Sum_{j=1..k} binomial(k,j)* (Sum_{i=j..n-k+j} (j^(i-j)/(i-j)! * binomial(k-j,n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j)) ) ), n>0. - Vladimir Kruchinin, Sep 10 2010
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(r^2/2 + n*r/(1+r)) * r^n * sqrt(r^2*(1+r)/n + 2+r-1/(1+r))), where r is the root of the equation r*(exp(r)*(1+r)-1-r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2)))/(2*LambertW(sqrt(n)/2)).
(End)
MATHEMATICA
nn = 30; a = x Exp[x]; Range[0, nn]! CoefficientList[Series[Exp[a - x^2/2! - x], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 10 2011 *)
CoefficientList[Series[Exp[-x - x^2/2 + x Exp[x]], {x, 0, 30}], x] Range[0, 30]! (* Eric W. Weisstein, Aug 10 2017 *)
Table[n! Sum[1/k! (Binomial[k, n-k] 2^(k-n) (-1)^k + Sum[Binomial[k, j] Sum[j^(i-j)/(i-j)! Binomial[k-j, n-i-k+j] 2^(i-j+k-n) (-1)^(k-j), {i, j, n-k+j}], {j, k}]), {k, n}], {n, 30}] (* Eric W. Weisstein, Aug 10 2017 *)
PROG
(Maxima) a(n):=n!*sum((binomial(k, n-k)*2^(k-n)*(-1)^k +sum(binomial(k, j) *sum(j^(i-j)/(i-j)!*binomial(k-j, n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j), i, j, n-k+j), j, 1, k))/k!, k, 1, n); /* Vladimir Kruchinin, Sep 10 2010 */
(PARI) x='x+O('x^30); Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x)))) \\ Altug Alkan, Aug 10 2017
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(-x -x^2/2 +x*Exp(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019
(Sage) m = 30; T = taylor(exp(-x -x^2/2 +x*exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 16 2000
STATUS
approved