${\displaystyle -\frac{18x^{10}-134x^9+446x^8-918x^7+1253x^6-1161x^5+736x^4-315x^3+87x^2-14x+1}{(x^2-3x+1){(2x-1)}^3{(-1+x)}^6}}$

1, 1, 2, 6, 21, 75, 259, 849, 2638, 7817, 22275, 61539, 166007, 439844, 1150070, ...

${\displaystyle (x^2-3x+1){(2x-1)}^3{(-1+x)}^{6}F\left(x\right)+18x^{10}-134x^9+446x^8-918x^7+1253x^6-1161x^5+736x^4-315x^3+87x^2-14x+1=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=21}$
${\displaystyle a\left(5\right)=75}$
${\displaystyle a\left(6\right)=259}$
${\displaystyle a\left(7\right)=849}$
${\displaystyle a\left(8\right)=2638}$
${\displaystyle a\left(9\right)=7817}$
${\displaystyle a\left(10\right)=22275}$
${\displaystyle a(n+5)=8a\left(n\right)-36a(n+1)+50a(n+2)-31a(n+3)+9a(n+4)-\frac{n(n^4-20n^3+55n^2+260n-176)}{120},n\ge 11}$

${\displaystyle \frac{(36\sqrt{5}+60){(\frac{3}{2}-\frac{\sqrt{5}}{2})}^{-n}}{120}+\frac{(-36\sqrt{5}+60){(\frac{3}{2}+\frac{\sqrt{5}}{2})}^{-n}}{120}+2^{n+1}+\frac{(30n^2-270n)2^n}{120}-\frac{n^5}{120}+\frac{n^4}{12}-\frac{n^3}{8}+\frac{11n^2}{12}-\frac{13n}{15}-2}$