${\displaystyle \frac{x^8-6x^7+39x^6-80x^5+97x^4-75x^3+35x^2-9x+1}{{(2x-1)}^2{(x-1)}^6}}$

1, 1, 2, 6, 20, 61, 164, 396, 883, 1865, 3811, 7662, 15346, 30869, 62622, ...

${\displaystyle {(2x-1)}^2{(x-1)}^{6}F\left(x\right)-x^8+6x^7-39x^6+80x^5-97x^4+75x^3-35x^2+9x-1=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=20}$
${\displaystyle a\left(5\right)=61}$
${\displaystyle a\left(6\right)=164}$
${\displaystyle a\left(7\right)=396}$
${\displaystyle a\left(8\right)=883}$
${\displaystyle a(n+2)=-4a\left(n\right)+4a(n+1)+\frac{(n-4)(3n^4-33n^3+23n^2-43n-30)}{120},n\ge 9}$

${\displaystyle \{\begin{array}{cc}1& n=0\\ \frac{n^5}{40}-\frac{n^4}{8}+\frac{7n^3}{24}-\frac{3n^2}{8}+\frac{11n}{60}+1+\frac{2^{n}n}{4}-\frac{2^n}{4}& \text{otherwise}\end{array}}$