A133297 a(n) = n! * Sum_{k=1..n} (-1)^(k + 1) * n^(n - k - 1)/(n - k)!.

0, 1, 1, 5, 34, 329, 4056, 60997, 1082320, 22137201, 512801920, 13269953861, 379400765184, 11877265764025, 404067857880064, 14843708906336325, 585606019079612416, 24693567694861202273, 1108343071153648926720, 52757597474618636748421, 2654611611461360017408000

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

E.g.f.: log(1 - LambertW(-x)).

a(n) ~ n^(n-1)/2. - Vaclav Kotesovec, Sep 25 2013

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

From Thomas Scheuerle, Nov 17 2023: (Start)

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)

a(n) = n^(n - 1) * hypergeom([1, 1-n], [], 1/n) for n > 0. - Peter Luschny, Mar 09 2026

Maple

a := n -> ifelse(n = 0, 0, n^(n-1)*hypergeom([1, 1-n], [], 1/n)): seq(simplify(a(n)), n = 0..20);  # Peter Luschny, Mar 09 2026

Mathematica

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 *)

Prog

(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]]) >; [a(n): n in [0..25]]; // G. C. Greubel, Aug 02 2019
(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))
[a(n) for n in (0..25)] # G. C. Greubel, Aug 02 2019
(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;
List([0..25], n-> a(n) ); # G. C. Greubel, Aug 02 2019
(Python)
from math import factorial
def a(n: int) -> int:
    if n == 0: return 0
    t = factorial(n - 1)
    if n % 2 == 0: t = -t
    f = t
    for k in range(n, 1, -1):
        t = -n * t // (n - k + 1)
        f += t
    return f
print([a(n) for n in range(0, 21)])  # Peter Luschny, Mar 09 2026

Crossrefs

Cf. A001865 (Gamma(n, n)/exp(-n)).

Cf. A000272, A000435, A277458, A292977.

Sequence in context: A357422 A197714 A198078 * A054931 A362771 A392999

Adjacent sequences: A133294 A133295 A133296 * A133298 A133299 A133300

Keyword

nonn,easy

Author

Vladeta Jovovic, Oct 17 2007

Extensions

More terms from Stefan Steinerberger, Oct 19 2007

Status

approved