cp's OEIS Frontend

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

%I A065440 #60 Aug 17 2025 20:18:27
%S A065440 1,0,1,8,81,1024,15625,279936,5764801,134217728,3486784401,
%T A065440 100000000000,3138428376721,106993205379072,3937376385699289,
%U A065440 155568095557812224,6568408355712890625,295147905179352825856,14063084452067724991009,708235345355337676357632
%N A065440 a(n) = (n-1)^n.
%C A065440 a(n) is the number of functions from {1,2,...,n} into {1,2,...,n} that have no fixed points.
%C A065440 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
%H A065440 Harry J. Smith, <a href="/A065440/b065440.txt">Table of n, a(n) for n = 0..100</a>
%H A065440 Mustafa Obaid et al., <a href="http://arxiv.org/abs/1307.7573">The number of complete exceptional sequences for a Dynkin algebra</a>, arXiv preprint arXiv:1307.7573 [math.RT], 2013.
%F A065440 a(n) = A007778(n-1).
%F A065440 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).
%F A065440 a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*n^(n-k). - _Robert FERREOL_, Mar 28 2017
%F A065440 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
%t A065440 Table[(n-1)^n,{n,0,20}] (* _Harvey P. Dale_, Jan 03 2015 *)
%o A065440 (PARI) a(n) = { (n - 1)^n } \\ _Harry J. Smith_, Oct 19 2009
%o A065440 (Magma)
%o A065440 A065440:= func< n | (n-1)^n >;
%o A065440 [A065440(n): n in [0..30]]; // _G. C. Greubel_, Mar 23 2025
%o A065440 (SageMath)
%o A065440 def A065440(n): return (n-1)**n
%o A065440 print([A065440(n) for n in range(31)]) # _G. C. Greubel_, Mar 23 2025
%Y A065440 Essentially the same as A007778 - note T(x) = -W(-x).
%Y A065440 Column k=0 of A055134.
%Y A065440 Row sums of A350452.
%Y A065440 Cf. A158497, A284458.
%K A065440 nonn,easy
%O A065440 0,4
%A A065440 _Len Smiley_, Nov 17 2001