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%I #52 Sep 08 2022 08:44:52
%S 1,2,9,32,625,324,117649,131072,4782969,1562500,25937424601,35831808,
%T 23298085122481,110730297608,4805419921875,562949953421312,
%U 48661191875666868481,91507169819844,104127350297911241532841,640000000000000000,865405750887126927009
%N Numerator of (n+1)^n/n!.
%C Also denominator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. - _N. J. A. Sloane_, Jul 31 2013
%H Vincenzo Librandi, <a href="/A036505/b036505.txt">Table of n, a(n) for n = 0..200</a>
%H Helmut Prodinger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i3p7">An identity conjectured by Lacasse via the tree function</a>, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
%F a(n) = A090878(n+1)/Sum_{k=0..n+1} (A128433(n+1)/A128434(n+1)). - _Reinhard Zumkeller_, Mar 03 2007
%F G.f.: -x*e^(-LambertW(-x))/((LambertW(-x)+1)*LambertW(-x)). - _Vladimir Kruchinin_, Feb 04 2013
%F A simpler g.f. is 1/(1 + LambertW(-x)). - _Jean-François Alcover_, Feb 04 2013
%p a:=n -> numer((n+1)^n/factorial(n)): A036505 := [seq(a(n), n=0..20)]; # _Muniru A Asiru_, Feb 12 2018
%t CoefficientList[Series[1/(1 + ProductLog[-x]), {x, 0, 21}], x] // Numerator // Rest (* _Jean-François Alcover_, Feb 04 2013, after _Vladimir Kruchinin_ *)
%o (Magma) [Numerator((n+1)^n/Factorial(n)): n in [0..20]]; // _Vincenzo Librandi_, Sep 10 2013
%o (GAP) List([0..20], n -> NumeratorRat((n+1)^n/Factorial(n))); # _Muniru A Asiru_, Feb 12 2018
%o (PARI) my(x='x+O('x^30)); apply(x -> numerator(x), Vec(-1+1/(1+lambertw(-x)))) \\ _G. C. Greubel_ and _Michel Marcus_, Feb 08 2019
%o (Sage) [numerator((n+1)^n/factorial(n)) for n in (0..20)] # _G. C. Greubel_, Feb 08 2019
%Y Cf. A036503, A063170.
%Y Cf. A095996 (denominators).
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_
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