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%I #58 Feb 20 2023 07:56:47
%S 1,2,7,43,393,4721,69853,1225757,24866481,572410513,14738647221,
%T 419682895325,13094075689225,444198818128313,16278315877572141,
%U 640854237634448101,26973655480577228769,1208724395795734172705,57453178877303382607717,2887169565412587866031533
%N Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).
%C Binomial transform of A000312. - _Tilman Neumann_, Dec 13 2008
%C a(n) is the number of partial functions on {1,2,...,n} that are endofunctions. See comments in A000169 and A126285 by _Franklin T. Adams-Watters_. - _Geoffrey Critzer_, Dec 19 2011
%H Winston de Greef, <a href="/A086331/b086331.txt">Table of n, a(n) for n = 0..385</a> (first 201 terms from Vincenzo Librandi)
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Interesting asymptotic formulas for binomial sums</a>, Jun 09 2013
%F a(n) = Sum_{k=0..n} binomial(n,k)*k^k.
%F a(n) ~ e^(1/e)*n^n * (1 + 1/(2*e*n)) ~ 1.444667861... * n^n. - _Vaclav Kotesovec_, Nov 27 2012
%F G.f.: Sum_{k>=0} (k * x)^k/(1 - x)^(k+1). - _Seiichi Manyama_, Jul 04 2022
%e a(2) = 7 because {}->{}, 1->1, 2->2, and the four functions from {1,2} into {1,2}. Note A000169(2) = 9 because it counts these 7 and 1->2, 2->1.
%p a:= n-> add(binomial(n,k)*k^k, k=0..n):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Dec 30 2021
%t nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[x]/(1-t),{x,0,nn}],x] (* _Geoffrey Critzer_, Dec 19 2011 *)
%o (PARI) a(n) = sum(k=0,n, binomial(n, k)*k^k ); \\ _Joerg Arndt_, May 10 2013
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-x)^(k+1))) \\ _Seiichi Manyama_, Jul 04 2022
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k*x)^k/k!))) \\ _Seiichi Manyama_, Jul 04 2022
%Y Cf. A069856, A204042, A277454, A277456, A323280.
%K nonn
%O 0,2
%A _Vladeta Jovovic_, Sep 01 2003
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