${\displaystyle \frac{x^8-6x^7-8x^6+31x^5-49x^4+48x^3-27x^2+8x-1}{{(2x-1)}^2{(x-1)}^5}}$

1, 1, 2, 6, 20, 61, 164, 400, 915, 2014, 4341, 9263, 19686, 41783, 88642, ...

${\displaystyle -{(2x-1)}^2{(x-1)}^{5}F\left(x\right)+x^8-6x^7-8x^6+31x^5-49x^4+48x^3-27x^2+8x-1=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=20}$
${\displaystyle a\left(5\right)=61}$
${\displaystyle a\left(6\right)=164}$
${\displaystyle a\left(7\right)=400}$
${\displaystyle a\left(8\right)=915}$
${\displaystyle a(n+2)=-4a\left(n\right)+4a(n+1)+\frac{(n-4)(3n^3-34n^2+41n-18)}{24},n\ge 9}$

${\displaystyle \{\begin{array}{cc}\frac{5}{4}& n=0\\ 1+\frac{32^{n}n}{8}+\frac{15n^2}{8}-\frac{25n}{12}-\frac{11n^3}{12}+\frac{n^4}{8}& \text{otherwise}\end{array}}$