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%I #51 Dec 23 2021 02:52:35
%S 0,1,4,24,224,2880,47232,942592,22171648,600698880,18422374400,
%T 630897721344,23864653578240,988197253808128,44460603225407488,
%U 2159714024218951680,112652924603290615808,6280048587936003784704,372616014329572403183616,23445082059018189741752320,1559275240299007139066675200
%N Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.
%C Essentially the same as A038049.
%C Also the number of rooted trees whose nodes are labeled with the blocks of a set partition of {1,2,...,n} each having a distinguished element. (See A000248.)
%C The bijection is straightforward. The trees correspond to functional digraphs mapping the distinguished elements towards the root. All the elements within each block are mapped to the distinguished element of that block. The distinguished element in the root node is the fixed point.
%H Vincenzo Librandi, <a href="/A216857/b216857.txt">Table of n, a(n) for n = 0..200</a>
%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%F E.g.f.: T(x*exp(x)) where T(x) is the e.g.f. for A000169.
%F a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-1).
%F a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n)*LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Jul 09 2013
%F O.g.f.: Sum_{n>=0} n^(n-1)* x^n / (1 - n*x)^(n+1). - _Paul D. Hanna_, May 22 2018
%F E.g.f.: the compositional inverse of A(x) is -A(-x). - _Alexander Burstein_, Aug 11 2018
%t With[{nmax = 20}, CoefficientList[Series[-LambertW[-x*Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* modified by _G. C. Greubel_, Nov 15 2017 *)
%o (PARI) for(n=0,30, print1(sum(k=1,n, binomial(n,k)*k^(n-1)), ", ")) \\ _G. C. Greubel_, Nov 15 2017
%o (PARI) my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-x*exp(x))))) \\ _G. C. Greubel_, Nov 15 2017
%Y Cf. A048802, A072034.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Sep 17 2012