cp's OEIS Frontend

A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

%I A216857 #51 Dec 23 2021 02:52:35
%S A216857 0,1,4,24,224,2880,47232,942592,22171648,600698880,18422374400,
%T A216857 630897721344,23864653578240,988197253808128,44460603225407488,
%U A216857 2159714024218951680,112652924603290615808,6280048587936003784704,372616014329572403183616,23445082059018189741752320,1559275240299007139066675200
%N A216857 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 A216857 Essentially the same as A038049.
%C A216857 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 A216857 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 A216857 Vincenzo Librandi, <a href="/A216857/b216857.txt">Table of n, a(n) for n = 0..200</a>
%H A216857 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 A216857 E.g.f.: T(x*exp(x)) where T(x) is the e.g.f. for A000169.
%F A216857 a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-1).
%F A216857 a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n)*LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Jul 09 2013
%F A216857 O.g.f.: Sum_{n>=0} n^(n-1)* x^n / (1 - n*x)^(n+1). - _Paul D. Hanna_, May 22 2018
%F A216857 E.g.f.: the compositional inverse of A(x) is -A(-x). - _Alexander Burstein_, Aug 11 2018
%t A216857 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 A216857 (PARI) for(n=0,30, print1(sum(k=1,n, binomial(n,k)*k^(n-1)), ", ")) \\ _G. C. Greubel_, Nov 15 2017
%o A216857 (PARI) my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-x*exp(x))))) \\ _G. C. Greubel_, Nov 15 2017
%Y A216857 Cf. A048802, A072034.
%K A216857 nonn
%O A216857 0,3
%A A216857 _Geoffrey Critzer_, Sep 17 2012