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 A053506 a(n) = (n-1)*n^(n-2). 24
 0, 1, 6, 48, 500, 6480, 100842, 1835008, 38263752, 900000000, 23579476910, 681091006464, 21505924728444, 737020860878848, 27246730957031250, 1080863910568919040, 45798768824157052688, 2064472028642102280192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 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 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 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 REFERENCES 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) R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2. LINKS G. C. Greubel, Table of n, a(n) for n = 1..250 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets Eric Weisstein's World of Mathematics, Graph Edge FORMULA E.g.f.: LambertW(-x)^2/2. - Vladeta Jovovic, Apr 07 2001 E.g.f. if offset 0: W(-x)^2/((1+W(-x))*x), W(x) Lambert's function (principal branch). 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 a(n) = Sum_{i=1..n-1} binomial(n-1,i-1)*i^(i-2)*(n-i)^(n-i). - Dmitry Kruchinin, Oct 28 2013 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 MATHEMATICA Table[(n-1)*n^(n-2), {n, 20}] PROG (PARI) vector(20, n, (n-1)*n^(n-2)) \\ G. C. Greubel, Jan 18 2017 (Magma) [(n-1)*n^(n-2): n in [1..20]]; // G. C. Greubel, May 15 2019 (Sage) [(n-1)*n^(n-2) for n in (1..20)] # G. C. Greubel, May 15 2019 (GAP) List([1..20], n-> (n-1)*n^(n-2)) # G. C. Greubel, May 15 2019 CROSSREFS Cf. A000169, A000312, A053507, A053508, A053509, A083483, A195242. Cf. A001865 which is the sum of A000169 + A053506 + A065513 + A065888 + ... Sequence in context: A052639 A248744 A261900 * A055861 A052567 A351340 Adjacent sequences:  A053503 A053504 A053505 * A053507 A053508 A053509 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 15 2000 STATUS approved

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Last modified September 26 08:43 EDT 2022. Contains 356993 sequences. (Running on oeis4.)