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A133297
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a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*n^(n-k-1)/(n-k)!.
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8
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0, 1, 1, 5, 34, 329, 4056, 60997, 1082320, 22137201, 512801920, 13269953861, 379400765184, 11877265764025, 404067857880064, 14843708906336325, 585606019079612416, 24693567694861202273, 1108343071153648926720, 52757597474618636748421, 2654611611461360017408000
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OFFSET
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0,4
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LINKS
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FORMULA
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E.g.f.: log(1-LambertW(-x)).
Conjecture: a(n) = (n-1)!*( Sum_{k >= 0} (-1)^k * n^(n+k)/(n+k)! - (-1/e)^n ) for n >= 1. Cf. A000435. - Peter Bala, Jul 23 2021
This conjecture is true. Let "gamma" be the lower incomplete gamma function: gamma(n, x) = (n-1)! (1 - exp(-x)*Sum_{k = 0..n-1} x^k/k! ), then we can get the upper incomplete gamma function Gamma(n, x) = gamma(n, oo) - gamma(n, x). By inserting according the formula below, we will obtain the formula from Peter Bala.
a(n) = (-1)^(n+1)*Gamma(n, -n)/exp(n) = (-1)^(n+1)*A292977(n-1, n), for n > 0, where Gamma is the upper incomplete gamma function. (End)
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MATHEMATICA
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Table[n!*Sum[(-1)^(k+1)*n^(n-k-1)/(n-k)!, {k, n}], {n, 0, 25}] (* Stefan Steinerberger, Oct 19 2007 *)
With[{m=25}, CoefficientList[Series[Log[1-LambertW[-x]], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Aug 02 2019 *)
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PROG
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(PARI) my(x='x+O('x^25)); concat([0], Vec(serlaplace( log(1-lambertw(-x)) ))) \\ G. C. Greubel, Aug 02 2019
(Magma)
a:= func< n | n eq 0 select 0 else Factorial(n)*(&+[(-1)^(k+1)*n^(n-k-1)/Factorial(n-k): k in [1..n]]) >;
(SageMath)
def a(n):
if (n==0): return 0
else: return factorial(n)*sum((-1)^(k+1)*n^(n-k-1)/factorial(n-k) for k in (1..n))
(GAP)
a:= function(n)
if n=0 then return 0;
else return Factorial(n)*Sum([1..n], k-> (-1)^(k+1)*n^(n-k-1)/Factorial(n-k));
fi;
end;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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