OFFSET
0,3
COMMENTS
a(n) = |non-injective functions [n]->[n]| = |non-surjective functions [n]->[n]|.
Fit a polynomial f of degree n-1 to the first n n-th powers of nonnegative integers. Then a(n) = f(n). - Franklin T. Adams-Watters, Dec 28 2006
n^n > n! for n >= 3. [Mitrinovic]
REFERENCES
D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 193, 3.1.22.
LINKS
T. D. Noe and Vincenzo Librandi, Table of n, a(n) for n = 0..300 [T. D. Noe computed terms 0-50, May 11 2007; Vincenzo Librandi computed the first 300 terms, Aug 22 2011]
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Corollary 2.3(iii).
FORMULA
E.g.f.: 1/(1-T(x))-1/(1-x) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Dec 10 2012
MATHEMATICA
Join[{0}, Table[n^n - n!, {n, 20}]] (* Harvey P. Dale, Oct 11 2011 *)
PROG
(Magma) [(n^n-Factorial(n)): n in [0..20] ]; // Vincenzo Librandi, Aug 22 2011
(PARI) a(n)=n^n-n! \\ Charles R Greathouse IV, Aug 22 2011
(Python)
from math import factorial
def a(n): return n**n - factorial(n)
print([a(n) for n in range(20)]) # Michael S. Branicky, Aug 10 2021
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved