 One has also 
where formula_46 is obtained from formula_47 by removing the minus signs.Let 
denote the sum under consideration for integer formula_49

Define the following exponential generating function with (initially) indeterminate formula_50
We find 
This is an entire function in formula_50 so that formula_50 can be taken to be any complex number.We next recall the exponential generating function for the Bernoulli polynomials formula_55 
where formula_57 denotes the Bernoulli number (with the convention formula_58).