${\displaystyle -\frac{x^{10}-3x^9+2x^8+4x^7-9x^6+24x^5-43x^4+38x^3-22x^2+7x-1}{{(x-1)}^8}}$

1, 1, 2, 6, 21, 74, 237, 668, 1667, 3750, 7743, 14898, 27033, 46698, 77369, ...

${\displaystyle {(x-1)}^{8}F\left(x\right)+x^{10}-3x^9+2x^8+4x^7-9x^6+24x^5-43x^4+38x^3-22x^2+7x-1=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=21}$
${\displaystyle a\left(5\right)=74}$
${\displaystyle a\left(6\right)=237}$
${\displaystyle a\left(7\right)=668}$
${\displaystyle a\left(8\right)=1667}$
${\displaystyle a\left(9\right)=3750}$
${\displaystyle a\left(10\right)=7743}$
${\displaystyle a\left(n\right)=15-\frac{869}{70}n+\frac{1}{120}{n}^6-\frac{23}{360}{n}^5+\frac{5}{24}{n}^4-\frac{47}{90}{n}^3+\frac{1}{2520}{n}^7+\frac{227}{60}{n}^2,n\ge 11}$

${\displaystyle \{\begin{array}{cc}1& n=0\text{ or }n=1\\ 2& n=2\\ 15-\frac{869}{70}n+\frac{1}{120}{n}^6-\frac{23}{360}{n}^5+\frac{5}{24}{n}^4-\frac{47}{90}{n}^3+\frac{1}{2520}{n}^7+\frac{227}{60}{n}^2& \text{otherwise}\end{array}}$