A072034 a(n) = Sum_{k=0..n} binomial(n,k)*k^n.

1, 1, 6, 54, 680, 11000, 217392, 5076400, 136761984, 4175432064, 142469423360, 5372711277824, 221903307604992, 9961821300640768, 482982946946734080, 25150966159083264000, 1400031335107317628928, 82960293298087664648192

Offset: 0

Author

Karol A. Penson, Jun 07 2002

Keywords

• nonn

Comments

The number of functions from {1,2,...,n} into a subset of {1,2,...,n} summed over all subsets. - Geoffrey Critzer, Sep 16 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

Crossrefs

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

Programs

Maple

seq(add(binomial(n,k)*k^n,k=0..n),n=0..17); # Peter Luschny, Jun 09 2015

Mathematica

Table[Sum[Binomial[n,k]k^n,{k,0,n}],{n,1,20}] (* Geoffrey Critzer, Sep 16 2012 *)

PARI

x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 10 2017

PARI

a(n) = sum(k=0, n, binomial(n,k)*k^n); \\ Michel Marcus, Nov 10 2017

Formula

E.g.f.: 1/(1+LambertW(-x*exp(x))). - Vladeta Jovovic, Mar 29 2008

a(n) ~ (n/(e*LambertW(1/e)))^n/sqrt(1+LambertW(1/e)). - Vaclav Kotesovec, Nov 26 2012

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

Extensions

Offset set to 0 and a(0) = 1 prepended by Peter Luschny, Jun 09 2015

E.g.f. edited to include a(0)=1 by Robert Israel, Jun 09 2015