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A055897
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a(n) = n*(n-1)^(n-1).
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16
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1, 2, 12, 108, 1280, 18750, 326592, 6588344, 150994944, 3874204890, 110000000000, 3423740047332, 115909305827328, 4240251492291542, 166680102383370240, 7006302246093750000, 313594649253062377472, 14890324713954061755186, 747581753430634213933056
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OFFSET
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1,2
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COMMENTS
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Total number of leaves in all labeled rooted trees with n nodes.
Number of endofunctions of [n] such that no element of [n-1] is fixed. E.g., a(3)=12: 123 -> 331, 332, 333, 311, 312, 313, 231, 232, 233, 211, 212, 213.
Number of functions f: {1, 2, ..., n} --> {1, 2, ..., n} such that f(1) != f(2), f(2) != f(3), ..., f(n-1) != f(n). - Warut Roonguthai, May 06 2006
Determinant of the n X n matrix ((2n, n^2, 0, ..., 0), (1, 2n, n^2, 0, ..., 0), (0, 1, 2n, n^2, 0, ..., 0), ..., (0, ..., 0, 1, 2n)). - Michel Lagneau, May 04 2010
Total number of points m such that f^(-1)(m) = {m}, (i.e., the preimage of m is the singleton set {m}) summed over all functions f:[n]->[n]. - Geoffrey Critzer, Jan 20 2022
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LINKS
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FORMULA
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E.g.f.: x/(1-T), where T=T(x) is Euler's tree function (see A000169).
a(n) = Sum_{k=1..n} A055302(n, k)*k.
a(n) = the n-th term of the (n-1)-th binomial transform of {1, 1, 4, 18, 96, ..., (n-1)*(n-1)!, ...} (cf. A001563). - Paul D. Hanna, Nov 17 2003
a(n) = (n-1)^(n-1) + Sum_{i=2..n} (n-1)^(n-i)*binomial(n-1, i-1)*(i-1) *(i-1)!. - Paul D. Hanna, Nov 17 2003
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MAPLE
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MATHEMATICA
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Join[{1}, Table[n(n-1)^(n-1), {n, 2, 20}]] (* Harvey P. Dale, Jul 18 2011 *)
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PROG
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(PARI) {a(n)=polcoeff(1/(1-n*x+x*O(x^n))^2, n)} \\ Paul D. Hanna, Dec 27 2012
(Haskell)
(Sage) [n*(n-1)^(n-1) for n in (1..20)] # G. C. Greubel, Aug 10 2019
(GAP) List([1..20], n-> n*(n-1)^(n-1)); # G. C. Greubel, Aug 10 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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