A065440 a(n) = (n-1)^n.

1, 0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632

Offset: 0

Author

Len Smiley, Nov 17 2001

Keywords

• nonn
• easy

Comments

a(n) is the number of functions from {1,2,...,n} into {1,2,...,n} that have no fixed points.

The probability that a random function from {1,2,...,n} into {1,2,...,n} has no fixed point is equal to a(n)/n^n; it tends to 1/e when n tends to infinity. - Robert FERREOL, Mar 29 2017

Links

• Harry J. Smith, Table of n, a(n) for n = 0..100
• Mustafa Obaid et al., The number of complete exceptional sequences for a Dynkin algebra, arXiv preprint arXiv:1307.7573 [math.RT], 2013.

Crossrefs

Essentially the same as A007778 - note T(x) = -W(-x).

Column k=0 of A055134.

Row sums of A350452.

Cf. A158497, A284458.

Programs

Magma

A065440:= func< n | (n-1)^n >;
[A065440(n): n in [0..30]]; // G. C. Greubel, Mar 23 2025

Mathematica

Table[(n-1)^n,{n,0,20}] (* Harvey P. Dale, Jan 03 2015 *)

PARI

a(n) = { (n - 1)^n } \\ Harry J. Smith, Oct 19 2009

SageMath

def A065440(n): return (n-1)**n
print([A065440(n) for n in range(31)]) # G. C. Greubel, Mar 23 2025

Formula

a(n) = A007778(n-1).

E.g.f.: x/(T(x)*(1-T(x))) (where T(x) is Euler's tree function, the E.g.f. for n^(n-1)) (see A000169).

a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*n^(n-k). - Robert FERREOL, Mar 28 2017

a(n) = Sum_{k=0..n} (-1)^k*binomial(n+2,k+2)*(k+1)*(2*k+n+3)^n. - Vladimir Kruchinin, Aug 13 2025