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

1, 2, 12, 108, 1280, 18750, 326592, 6588344, 150994944, 3874204890, 110000000000, 3423740047332, 115909305827328, 4240251492291542, 166680102383370240, 7006302246093750000, 313594649253062377472, 14890324713954061755186, 747581753430634213933056

Offset: 1

Author

Christian G. Bower, Jun 12 2000

Keywords

• nonn

Comments

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

For n > 1: a(n) = A240993(n-1) / A240993(n-2). - Reinhard Zumkeller, Aug 31 2014

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Frank Ellermann, Illustration of binomial transforms
• Index entries for sequences related to rooted trees

Crossrefs

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

Cf. A240993. Diagonal of A265583.

Programs

GAP

List([1..20], n-> n*(n-1)^(n-1)); # G. C. Greubel, Aug 10 2019

Haskell

a055897 n = n * (n - 1) ^ (n - 1) -- Reinhard Zumkeller, Aug 31 2014

Magma

[n*(n-1)^(n-1): n in [1..20]] // Wesley Ivan Hurt, Jun 26 2014

Maple

A055897:=n->`if`(n=1,1,n*(n-1)^(n-1)); seq(A055897(n), n=1..20); # Wesley Ivan Hurt, Jun 26 2014

Mathematica

Join[{1},Table[n(n-1)^(n-1), {n,2,20}]] (* Harvey P. Dale, Jul 18 2011 *)

PARI

{a(n)=polcoeff(1/(1-n*x+x*O(x^n))^2, n)} \\ Paul D. Hanna, Dec 27 2012

Sage

[n*(n-1)^(n-1) for n in (1..20)] # G. C. Greubel, Aug 10 2019

Formula

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

a(n) = [x^(n-1)] 1/(1 - (n-1)*x)^2. - Paul D. Hanna, Dec 27 2012

a(n) ~ exp(-1) * n^n. - Vaclav Kotesovec, Nov 14 2014

Extensions

Additional comments from Vladeta Jovovic, Mar 31 2001 and Len Smiley, Dec 11 2001