A036679 a(n) = n^n - n!.

0, 0, 2, 21, 232, 3005, 45936, 818503, 16736896, 387057609, 9996371200, 285271753811, 8915621446656, 302868879571453, 11111919647266816, 437892582706491375, 18446723150919663616, 827239906198908668177, 39346401672922831847424, 1978419534015213180291979

Offset: 0

Author

N. J. A. Sloane, G. L. Honaker, Jr.

Keywords

• nonn
• easy
• nice

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).

Crossrefs

Cf. A126130, diagonal of A101030.

Programs

Magma

[(n^n-Factorial(n)): n in [0..20] ]; // Vincenzo Librandi, Aug 22 2011

Mathematica

Join[{0}, Table[n^n - n!, {n, 20}]] (* Harvey P. Dale, Oct 11 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

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