A133297 - OEIS

%I #52 Jan 20 2024 09:41:20

%S 0,1,1,5,34,329,4056,60997,1082320,22137201,512801920,13269953861,

%T 379400765184,11877265764025,404067857880064,14843708906336325,

%U 585606019079612416,24693567694861202273,1108343071153648926720,52757597474618636748421,2654611611461360017408000

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

%H Alois P. Heinz, <a href="/A133297/b133297.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) ~ n^(n-1)/2. - _Vaclav Kotesovec_, Sep 25 2013

%F 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

%F From _Thomas Scheuerle_, Nov 17 2023: (Start)

%F 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.

%F 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)

%t Table[n!*Sum[(-1)^(k+1)*n^(n-k-1)/(n-k)!, {k,n}], {n,0,25}] (* _Stefan Steinerberger_, Oct 19 2007 *)

%t With[{m=25}, CoefficientList[Series[Log[1-LambertW[-x]], {x,0,m}], x]*Range[0,m]!] (* _G. C. Greubel_, Aug 02 2019 *)

%o (PARI) my(x='x+O('x^25)); concat([0], Vec(serlaplace( log(1-lambertw(-x)) ))) \\ _G. C. Greubel_, Aug 02 2019

%o (Magma)

%o 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]]) >;

%o [a(n): n in [0..25]]; // _G. C. Greubel_, Aug 02 2019

%o (SageMath)

%o def a(n):

%o     if (n==0): return 0

%o     else: return factorial(n)*sum((-1)^(k+1)*n^(n-k-1)/factorial(n-k) for k in (1..n))

%o [a(n) for n in (0..25)] # _G. C. Greubel_, Aug 02 2019

%o (GAP)

%o a:= function(n)

%o     if n=0 then return 0;

%o     else return Factorial(n)*Sum([1..n], k-> (-1)^(k+1)*n^(n-k-1)/Factorial(n-k));

%o     fi;

%o   end;

%o List([0..25], n-> a(n) ); # _G. C. Greubel_, Aug 02 2019

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

%Y Cf. A000272, A000435, A277458, A292977.

%K nonn,easy

%O 0,4

%A _Vladeta Jovovic_, Oct 17 2007

%E More terms from _Stefan Steinerberger_, Oct 19 2007