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
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013
Programs
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Maple
seq(add(binomial(n,k)*k^n,k=0..n),n=0..17); # Peter Luschny, Jun 09 2015
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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
Comments