Walking into an absolute sum
  We investigate a combinatorial sum that can be interpreted as the moments of a random variate, measuring the absolute distance to the origin in a symmetric Bernoulli random walk. These sums can be characterized by polynomials related to the Dumont-Foata polynomials. The sums corresponding to the odd moments have a connection to the Gandhi polynomials and Genocchi numbers. 
Hans J. H. Tuenter