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%I A053530 #52 Feb 19 2025 03:31:27
%S A053530 1,0,1,3,7,35,171,847,5041,32643,223705,1659581,13182159,110802133,
%T A053530 984241363,9212696235,90477239521,929604133343,9969157068273,
%U A053530 111329454692485,1291932988047775,15550838026589061,193833398512358011,2498039016973836491
%N A053530 Expansion of e.g.f.: exp(-x - x^2/2 + x*exp(x)).
%C A053530 The number of simple labeled graphs on n nodes whose connected components are stars. - _Geoffrey Critzer_, Dec 10 2011
%C A053530 Equivalently, the number of minimal edge covers of the complete graph K_n. - _Andrew Howroyd_, Aug 04 2017
%D A053530 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(b).
%H A053530 Vincenzo Librandi, <a href="/A053530/b053530.txt">Table of n, a(n) for n = 0..200</a>
%H A053530 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%H A053530 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%H A053530 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H A053530 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalEdgeCover.html">Minimal Edge Cover</a>
%F A053530 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
%F A053530 From _Vaclav Kotesovec_, Aug 06 2014: (Start)
%F A053530 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.
%F A053530 (a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2)))/(2*LambertW(sqrt(n)/2)).
%F A053530 (End)
%t A053530 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 *)
%t A053530 CoefficientList[Series[Exp[-x - x^2/2 + x Exp[x]], {x, 0, 30}], x] Range[0, 30]! (* _Eric W. Weisstein_, Aug 10 2017 *)
%t A053530 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 *)
%o A053530 (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 */
%o A053530 (PARI) x='x+O('x^30); Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x)))) \\ _Altug Alkan_, Aug 10 2017
%o A053530 (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
%o A053530 (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
%Y A053530 Cf. A000248, A210655.
%K A053530 nonn,changed
%O A053530 0,4
%A A053530 _N. J. A. Sloane_, Jan 16 2000

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