[seqfan] Generalization of a LambertW Identity

[Paul D Hanna]

SeqFans, 
      Consider the LambertW identity: 
 
1 = Sum_{n>=0} m*(n+m)^(n-1) * exp(-(n+m)*x) * x^n/n!.
 
I was surprised to find that the g.f.: 
 
Sum_{n>=0} n^n * m*(n+m)^(n-1) * exp(-n*(n+m)*x) * x^n/n! 
 
simplifies to a power series in x with integer coefficients for all integer m.  
 
Examples: 
https://oeis.org/A217900
https://oeis.org/A217901
https://oeis.org/A217902
 
More generally, let m, s>=0 and t be integers, then the series
 
Sum_{n>=0} n^(s*n) * m*(n*t+m)^(n-1) * exp(-n^s*(n*t+m)*x) * x^n/n! 
 
also simplifies to a power series in x with integer coefficients. 
  
The above statement is equivalent to showing that 
 
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^(s*n) * m*(k*t+m)^(n-1). 
 
is integral for n>=0 under the same conditions. 
 
There must be combinatorial interpretations for these sequences, but that is beyond my reach. 
      Paul