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 A006153 E.g.f.: 1/(1-x*exp(x)). (Formerly M3578) 21
 1, 1, 4, 21, 148, 1305, 13806, 170401, 2403640, 38143377, 672552730, 13044463641, 276003553860, 6326524990825, 156171026562838, 4130464801497105, 116526877671782896, 3492868475952497313, 110856698175372359346, 3713836169709782989993 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Without the first "1" = eigensequence of triangle A003566. - Gary W. Adamson, Dec 29 2008 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 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 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 REFERENCES S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22. Getu, S.; Shapiro, L. W.; Combinatorial view of the composition of functions. Ars Combin. 10 (1980), 131-145. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 110 Dennis Walsh, Assigning people into labeled groups with leaders FORMULA a(n) = n! * Sum_{k=0..n}(n-k)^k/k!. a(n) = Sum_{k=0..n} k! k^(n-k) binomial(n,k). 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 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 E.g.f. for a(n), n>=1: x*e^x/(1-x*e^x). - Dennis P. Walsh, Nov 15 2011 a(n) ~ n! / ((1+LambertW(1))*LambertW(1)^n). - Vaclav Kotesovec, Jun 21 2013\ O.g.f.: Sum_{n>=0} n! * x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018 EXAMPLE 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. a(4) = 148 = 4 + 48 + 72 + 24. MAPLE a := proc(n) local k; add(k^(n-k)*n!/(n-k)!, k=1..n); end; # for n >= 1 MATHEMATICA With[{nn=20}, CoefficientList[Series[1/(1-x Exp[x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Aug 29 2012 *) a[ n_] := If[n < 0, 0, n! + n! Sum[(n - k)^k / k!, {k, n}]]; (* Michael Somos, Jan 21 2019 *) PROG (PARI) x='x+O('x^66); /* that many terms */ egf=1/(1-x*exp(x)); /* = 1 + x + 2*x^2 + 7/2*x^3 + 37/6*x^4 + 87/8*x^5 +... */ Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */ (PARI) {a(n) = if(n<0, 0, n! * sum(k=0, n, (n-k)^k / k!))}; /* Michael Somos, Jan 21 2019 */ (Sage) def A006153_list(len):     f, R, C = 1, , +*(len-1)     for n in (1..len-1):         f *= n         for k in range(n, 0, -1):             C[k] = -C[k-1]*(1/(k-1) if k>1 else 1)         C = sum((-1)^k*C[k] for k in (1..n))         R.append(C*f)     return R print A006153_list(20) # Peter Luschny, Feb 21 2016 CROSSREFS Row sums of triangle A199673. Cf. A003566, A072597, A089148. Sequence in context: A307525 A163861 A247054 * A286286 A277505 A183387 Adjacent sequences:  A006150 A006151 A006152 * A006154 A006155 A006156 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Definition corrected by Joerg Arndt, Apr 30 2011 STATUS approved

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Last modified August 19 18:51 EDT 2019. Contains 326133 sequences. (Running on oeis4.)