OFFSET
1,2
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
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
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
MAPLE
MATHEMATICA
Join[{1}, Table[n(n-1)^(n-1), {n, 2, 20}]] (* Harvey P. Dale, Jul 18 2011 *)
PROG
(PARI) {a(n)=polcoeff(1/(1-n*x+x*O(x^n))^2, n)} \\ Paul D. Hanna, Dec 27 2012
(Magma) [n*(n-1)^(n-1): n in [1..20]] // Wesley Ivan Hurt, Jun 26 2014
(Haskell)
a055897 n = n * (n - 1) ^ (n - 1) -- Reinhard Zumkeller, Aug 31 2014
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian G. Bower, Jun 12 2000
EXTENSIONS
Additional comments from Vladeta Jovovic, Mar 31 2001 and Len Smiley, Dec 11 2001
STATUS
approved