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%I #60 Mar 23 2023 19:06:43
%S 0,1,6,48,500,6480,100842,1835008,38263752,900000000,23579476910,
%T 681091006464,21505924728444,737020860878848,27246730957031250,
%U 1080863910568919040,45798768824157052688,2064472028642102280192,98646963440126439346902,4980736000000000000000000
%N a(n) = (n-1)*n^(n-2).
%C a(n) is the number of endofunctions f of [n] which interchange a pair a<->b and for all x in [n] some iterate f^k(x) = a. E.g., a(3) = 6: 1<->2<-3; 3->1<->2; 2<->3<-1; 1->2<->3; 1<->3<-2; 2->1<->3. - _Len Smiley_, Nov 27 2001
%C If offset is 0: right side of the binomial sum n-> sum( i^(i-1) * (n-i+1)^(n-i)*binomial(n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
%C a(n) is the number of birooted labeled trees on n nodes in which the two root nodes are adjacent. - _N. J. A. Sloane_, May 01 2018
%C a(n) is the number of ways to partition the complete graph K_n into two components and choose an arborescence on each component. - _Harry Richman_, May 11 2022
%D A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.36)
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.
%H G. C. Greubel, <a href="/A053506/b053506.txt">Table of n, a(n) for n = 1..250</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphEdge.html">Graph Edge</a>
%F E.g.f.: LambertW(-x)^2/2. - _Vladeta Jovovic_, Apr 07 2001
%F E.g.f. if offset 0: W(-x)^2/((1+W(-x))*x), W(x) Lambert's function (principal branch).
%F The sequence 1, 1, 6, 48, ... satisfies a(n) = (n*(n+1)^n + 0^n)/(n+1); it is the main diagonal of A085388. - _Paul Barry_, Jun 30 2003
%F a(n) = Sum_{i=1..n-1} binomial(n-1,i-1)*i^(i-2)*(n-i)^(n-i). - _Dmitry Kruchinin_, Oct 28 2013
%F If offset = 0 and a(0) = 1 then a(n) = Sum_{k=0..n} (-1)^(n-k)* binomial(-k,-n)*n^k (cf. A195242). - _Peter Luschny_, Apr 11 2016
%t Table[(n-1)*n^(n-2), {n,20}]
%o (PARI) vector(20, n, (n-1)*n^(n-2)) \\ _G. C. Greubel_, Jan 18 2017
%o (Magma) [(n-1)*n^(n-2): n in [1..20]]; // _G. C. Greubel_, May 15 2019
%o (Sage) [(n-1)*n^(n-2) for n in (1..20)] # _G. C. Greubel_, May 15 2019
%o (GAP) List([1..20], n-> (n-1)*n^(n-2)) # _G. C. Greubel_, May 15 2019
%Y Cf. A000169, A000312, A053507, A053508, A053509, A083483, A195242.
%Y Cf. A001865 which is the sum of A000169 + A053506 + A065513 + A065888 + ...
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Jan 15 2000
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