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A349731 a(n) = -(-n)^n * FallingFactorial(1/n, n) for n >= 1 and a(0) = -1.

%I A349731 #29 Mar 01 2022 01:26:03
%S A349731 -1,1,1,10,231,9576,623645,58715280,7547514975,1270453824640,
%T A349731 271252029133449,71635824470246400,22929813173612997575,
%U A349731 8747686347650933760000,3921812703436118765113125,2041590849971133677650610176,1221367737152989777782325269375,832163138229382457228044554240000
%N A349731 a(n) = -(-n)^n * FallingFactorial(1/n, n) for n >= 1 and a(0) = -1.
%H A349731 G. C. Greubel, <a href="/A349731/b349731.txt">Table of n, a(n) for n = 0..200</a>
%F A349731 a(n) = -(-1)^n*Sum_{k=0..n}[n, n-k]*(-n)^k, where [n, k] denotes the Stirling cycle numbers A132393(n, k).
%p A349731 A349731 := n -> -add((-1)^(n-k)*Stirling1(n, n-k)*(-n)^k, k = 0..n):
%p A349731 seq(A349731(n), n = 0..17);
%t A349731 a[0] = -1; a[n_] := -(-n)^n * FactorialPower[1/n, n]; Array[a, 18, 0] (* _Amiram Eldar_, Dec 21 2021 *)
%o A349731 (SageMath)
%o A349731 def a(n): return -(-n)^n*falling_factorial(1/n, n) if n > 0 else -1
%o A349731 print([a(n) for n in (1..17)])
%o A349731 (Python)
%o A349731 from sympy import ff
%o A349731 from fractions import Fraction
%o A349731 def A349731(n): return -1 if n == 0 else -(-n)**n*ff(Fraction(1,n),n) # _Chai Wah Wu_, Dec 21 2021
%o A349731 (Magma) [-1,1] cat [Round(n^(n-1)*Gamma((n^2-1)/n)/Gamma((n-1)/n)): n in [2..30]]; // _G. C. Greubel_, Feb 22 2022
%Y A349731 The main diagonal of A349971 for n >= 1.
%Y A349731 The Stirling set counterpart is A318183.
%Y A349731 Cf. A008279, A092985, A132393.
%K A349731 sign
%O A349731 0,4
%A A349731 _Peter Luschny_, Dec 21 2021