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A055897 a(n) = n*(n-1)^(n-1). 10

%I

%S 1,2,12,108,1280,18750,326592,6588344,150994944,3874204890,

%T 110000000000,3423740047332,115909305827328,4240251492291542,

%U 166680102383370240,7006302246093750000,313594649253062377472,14890324713954061755186,747581753430634213933056

%N a(n) = n*(n-1)^(n-1).

%C Total number of leaves in all labeled rooted trees with n nodes.

%C 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.

%C 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

%C 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

%C For n > 1: a(n) = A240993(n-1) / A240993(n-2). - _Reinhard Zumkeller_, Aug 31 2014

%H Vincenzo Librandi, <a href="/A055897/b055897.txt">Table of n, a(n) for n = 1..200</a>

%H F. Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F E.g.f.: x/(1-T), where T=T(x) is Euler's tree function (see A000169).

%F a(n) = Sum_{k=1..n} A055302(n, k)*k.

%F 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

%F 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

%F a(n) = [x^(n-1)] 1/(1 - (n-1)*x)^2. - _Paul D. Hanna_, Dec 27 2012

%F a(n) ~ exp(-1) * n^n. - _Vaclav Kotesovec_, Nov 14 2014

%p A055897:=n->`if`(n=1,1,n*(n-1)^(n-1)); seq(A055897(n), n=1..20); # _Wesley Ivan Hurt_, Jun 26 2014

%t Join[{1},Table[n(n-1)^(n-1), {n,2,20}]] (* _Harvey P. Dale_, Jul 18 2011 *)

%o (PARI) {a(n)=polcoeff(1/(1-n*x+x*O(x^n))^2, n)} \\ _Paul D. Hanna_, Dec 27 2012

%o (MAGMA) [n*(n-1)^(n-1): n in [1..20]] // _Wesley Ivan Hurt_, Jun 26 2014

%o (Haskell)

%o a055897 n = n * (n - 1) ^ (n - 1) -- _Reinhard Zumkeller_, Aug 31 2014

%Y Cf. A003227, A003228, A055314, A055540, A055541, A060226, A118537.

%Y Cf. A240993. Diagonal of A265583.

%K nonn

%O 1,2

%A _Christian G. Bower_, Jun 12 2000

%E Additional comments from _Vladeta Jovovic_, Mar 31 2001 and _Len Smiley_, Dec 11 2001

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Last modified July 18 19:00 EDT 2019. Contains 325144 sequences. (Running on oeis4.)