Mott polynomials

In mathematics the Mott polynomials "s""n"("x") are polynomials introduced by who applied them to a problem in the theory of electrons. 
They are given by the exponential generating function 
Because the factor in the exponential has the power series
in terms of Catalan numbers formula_3, the coefficient in front of formula_4 of the polynomial can be written as
according to the general formula for generalized Appell polynomials,
where the sum is over all compositions formula_6 of formula_7 into formula_8 positive odd integers. The empty product appearing for formula_9 equals 1. Special values, where all contributing Catalan numbers equal 1, are
By differentiation the recurrence for the first derivative becomes
The first few of them are 
The polynomials "s""n"("x") form the associated Sheffer sequence for –2"t"/(1–t2) .
 give an explicit expression for them in terms of the generalized hypergeometric function 3F0: