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

-1, 1, 1, 10, 231, 9576, 623645, 58715280, 7547514975, 1270453824640, 271252029133449, 71635824470246400, 22929813173612997575, 8747686347650933760000, 3921812703436118765113125, 2041590849971133677650610176, 1221367737152989777782325269375, 832163138229382457228044554240000

Offset: 0

Author

Peter Luschny, Dec 21 2021

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200

Crossrefs

The main diagonal of A349971 for n >= 1.

The Stirling set counterpart is A318183.

Cf. A008279, A092985, A132393.

Programs

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

Maple

A349731 := n -> -add((-1)^(n-k)*Stirling1(n, n-k)*(-n)^k, k = 0..n):
seq(A349731(n), n = 0..17);

Mathematica

a[0] = -1; a[n_] := -(-n)^n * FactorialPower[1/n, n]; Array[a, 18, 0] (* Amiram Eldar, Dec 21 2021 *)

Python

from sympy import ff
from fractions import Fraction
def A349731(n): return -1 if n == 0 else -(-n)**n*ff(Fraction(1,n),n) # Chai Wah Wu, Dec 21 2021

SageMath

def a(n): return -(-n)^n*falling_factorial(1/n, n) if n > 0 else -1
print([a(n) for n in (1..17)])

Formula

a(n) = -(-1)^n*Sum_{k=0..n}[n, n-k]*(-n)^k, where [n, k] denotes the Stirling cycle numbers A132393(n, k).