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 A006153 E.g.f.: 1/(1-x*exp(x)). (Formerly M3578) 21

%I M3578

%S 1,1,4,21,148,1305,13806,170401,2403640,38143377,672552730,

%T 13044463641,276003553860,6326524990825,156171026562838,

%U 4130464801497105,116526877671782896,3492868475952497313,110856698175372359346,3713836169709782989993

%N E.g.f.: 1/(1-x*exp(x)).

%C Without the first "1" = eigensequence of triangle A003566. - _Gary W. Adamson_, Dec 29 2008

%C a(n) is the sum of the row entries of triangle A199673, that is, a(n) is the number of ways to assign n people into labeled groups and then to assign a leader for each group from its members; see example below. - _Dennis P. Walsh_, Nov 15 2011

%C a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that for some j>1, f^j=f where f^j denotes iterated functional composition. Equivalently, the number of endofunctions such that every element is mapped to a recurrent element. Equivalently, every vertex of the functional digraph is at a distance at most 1 from a cycle. - _Geoffrey Critzer_, Jan 21 2012

%C Numerators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." - _Michael Somos_, Jan 21 2019

%D S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

%D Getu, S.; Shapiro, L. W.; Combinatorial view of the composition of functions. Ars Combin. 10 (1980), 131-145.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).

%H Alois P. Heinz, <a href="/A006153/b006153.txt">Table of n, a(n) for n = 0..200</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=110">Encyclopedia of Combinatorial Structures 110</a>

%H Dennis Walsh, <a href="http://frank.mtsu.edu/~dwalsh/GROUPCNT.pdf"> Assigning people into labeled groups with leaders</a>

%F a(n) = n! * Sum_{k=0..n}(n-k)^k/k!.

%F a(n) = Sum_{k=0..n} k! k^(n-k) binomial(n,k).

%F For n>=1, a(n-1) = b(n) where b(1)=1 and b(n) = Sum_{i=1..n-1} i*binomial(n-1, i)*b(i). - _Benoit Cloitre_, Nov 13 2004

%F a(n) = Sum_{k=1..n}A199673(n,k) = Sum_{k=1..n}n! k^(n-k)/(n-k)!. - _Dennis P. Walsh_, Nov 15 2011

%F E.g.f. for a(n), n>=1: x*e^x/(1-x*e^x). - _Dennis P. Walsh_, Nov 15 2011

%F a(n) ~ n! / ((1+LambertW(1))*LambertW(1)^n). - _Vaclav Kotesovec_, Jun 21 2013\

%F O.g.f.: Sum_{n>=0} n! * x^n / (1 - n*x)^(n+1). - _Paul D. Hanna_, May 22 2018

%e a(3) = 21 since there are 21 ways to assign 3 people into labeled groups with designated leaders. If there is one group, there are 3 ways to select a leader from the 3 people in the group. If there are two groups (group 1 and group 2), there are 6 ways to assign leaders and then 2 ways to select a group for the remaining person, and thus there are 12 assignments. If there are three groups (group1, group 2, and group3), each person is a leader of their singleton group, and there are 6 ways to assign the 3 people to the 3 groups. Hence a(3) = 3 + 12 + 6 = 21.

%e a(4) = 148 = 4 + 48 + 72 + 24.

%p a := proc(n) local k; add(k^(n-k)*n!/(n-k)!,k=1..n); end; # for n >= 1

%t With[{nn=20},CoefficientList[Series[1/(1-x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 29 2012 *)

%t a[ n_] := If[n < 0, 0, n! + n! Sum[(n - k)^k / k!, {k, n}]]; (* _Michael Somos_, Jan 21 2019 *)

%o (PARI) x='x+O('x^66); /* that many terms */

%o egf=1/(1-x*exp(x)); /* = 1 + x + 2*x^2 + 7/2*x^3 + 37/6*x^4 + 87/8*x^5 +... */

%o Vec(serlaplace(egf)) /* show terms */ /* _Joerg Arndt_, Apr 30 2011 */

%o (PARI) {a(n) = if(n<0, 0, n! * sum(k=0, n, (n-k)^k / k!))}; /* _Michael Somos_, Jan 21 2019 */

%o (Sage)

%o def A006153_list(len):

%o f, R, C = 1, , +*(len-1)

%o for n in (1..len-1):

%o f *= n

%o for k in range(n, 0, -1):

%o C[k] = -C[k-1]*(1/(k-1) if k>1 else 1)

%o C = sum((-1)^k*C[k] for k in (1..n))

%o R.append(C*f)

%o return R

%o print A006153_list(20) # _Peter Luschny_, Feb 21 2016

%Y Row sums of triangle A199673.

%Y Cf. A003566, A072597, A089148.

%K nonn,easy,nice

%O 0,3

%A _Simon Plouffe_ and _N. J. A. Sloane_

%E Definition corrected by _Joerg Arndt_, Apr 30 2011

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Last modified November 13 15:41 EST 2019. Contains 329106 sequences. (Running on oeis4.)