%I #46 Feb 20 2021 11:03:30

%S 1,1,6,54,680,11000,217392,5076400,136761984,4175432064,142469423360,

%T 5372711277824,221903307604992,9961821300640768,482982946946734080,

%U 25150966159083264000,1400031335107317628928,82960293298087664648192

%N a(n) = Sum_{k=0..n} binomial(n,k)*k^n.

%C The number of functions from {1,2,...,n} into a subset of {1,2,...,n} summed over all subsets. - _Geoffrey Critzer_, Sep 16 2012

%H Vincenzo Librandi, <a href="/A072034/b072034.txt">Table of n, a(n) for n = 0..200</a>

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Interesting asymptotic formulas for binomial sums</a>, Jun 09 2013

%F E.g.f.: 1/(1+LambertW(-x*exp(x))). - _Vladeta Jovovic_, Mar 29 2008

%F a(n) ~ (n/(e*LambertW(1/e)))^n/sqrt(1+LambertW(1/e)). - _Vaclav Kotesovec_, Nov 26 2012

%F O.g.f.: Sum_{n>=0} n^n * x^n / (1 - n*x)^(n+1). - _Paul D. Hanna_, May 22 2018

%p seq(add(binomial(n,k)*k^n,k=0..n),n=0..17); # _Peter Luschny_, Jun 09 2015

%t Table[Sum[Binomial[n,k]k^n,{k,0,n}],{n,1,20}] (* _Geoffrey Critzer_, Sep 16 2012 *)

%o (PARI) x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(-x*exp(x))))) \\ _G. C. Greubel_, Nov 10 2017

%o (PARI) a(n) = sum(k=0, n, binomial(n,k)*k^n); \\ _Michel Marcus_, Nov 10 2017

%Y Cf. A088789, A242446, A256016, A336828, A341815.

%K nonn

%O 0,3

%A _Karol A. Penson_, Jun 07 2002

%E Offset set to 0 and a(0) = 1 prepended by _Peter Luschny_, Jun 09 2015

%E E.g.f. edited to include a(0)=1 by _Robert Israel_, Jun 09 2015