# Abel polynomials

The Abel polynomials in mathematics form a polynomial sequence, the nth term of which is of the form

${\displaystyle p_{n}(x)=x(x-an)^{n-1}.}$

The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician.

This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may be obtained from the Abel sequence in the umbral calculus.

## Examples

For a=1, the polynomials are (sequence A137452 in the OEIS)

${\displaystyle p_{0}(x)=1;}$
${\displaystyle p_{1}(x)=x;}$
${\displaystyle p_{2}(x)=-2x+x^{2};}$
${\displaystyle p_{3}(x)=9x-6x^{2}+x^{3};}$
${\displaystyle p_{4}(x)=-64x+48x^{2}-12x^{3}+x^{4};}$

For a=2, the polynomials are

${\displaystyle p_{0}(x)=1;}$
${\displaystyle p_{1}(x)=x;}$
${\displaystyle p_{2}(x)=-4x+x^{2};}$
${\displaystyle p_{3}(x)=36x-12x^{2}+x^{3};}$
${\displaystyle p_{4}(x)=-512x+192x^{2}-24x^{3}+x^{4};}$
${\displaystyle p_{5}(x)=10000x-4000x^{2}+600x^{3}-40x^{4}+x^{5};}$
${\displaystyle p_{6}(x)=-248832x+103680x^{2}-17280x^{3}+1440x^{4}-60x^{5}+x^{6};}$