The Abel polynomials are a sequence of polynomials named after Niels Henrik Abel, defined by the following equation:
They occur, for instance, in the enumeration of trees (specifically, rooted labeled forests) as well as in connection with geometric probability (the random placement of nonoverlapping arcs on a circle). This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may be obtained from the Abel sequence using umbral calculus.
For , the polynomials are
For , the polynomials are