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A053530 E.g.f.: exp(-x-1/2*x^2+x*exp(x)). 7
1, 0, 1, 3, 7, 35, 171, 847, 5041, 32643, 223705, 1659581, 13182159, 110802133, 984241363, 9212696235, 90477239521, 929604133343, 9969157068273, 111329454692485, 1291932988047775, 15550838026589061, 193833398512358011, 2498039016973836491 (list; graph; refs; listen; history; text; internal format)
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((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), 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 = 16; 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, 20}], x] Range[0, 20]! (* 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, 20}] (* 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^99); Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x)))) \\ Altug Alkan, Aug 10 2017

CROSSREFS

Cf. A000248, A210655.

Sequence in context: A121130 A006099 A240272 * A215575 A266049 A132102

Adjacent sequences:  A053527 A053528 A053529 * A053531 A053532 A053533

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 16 2000

STATUS

approved

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Last modified October 21 06:25 EDT 2018. Contains 316405 sequences. (Running on oeis4.)