${\displaystyle \frac{(x^2-3x+1)(x^5-2x^4-3x^3+6x^2-4x+1)\sqrt{1-4x}-9x^7+19x^6+10x^5-59x^4+63x^3-33x^2+9x-1}{2x^2(x^2+x-1)(x^2-3x+1){(x-1)}^2}}$

1, 1, 2, 6, 21, 76, 275, 989, 3544, 12696, 45578, 164194, 593966, 2158090, 7875503, ...

${\displaystyle x^2{(x^2-3x+1)}^2{(x^2+x-1)}^2{(x-1)}^{4}F{\left(x\right)}^2+(x^2-3x+1)(x^2+x-1)(9x^7-19x^6-10x^5+59x^4-63x^3+33x^2-9x+1){(x-1)}^{2}F\left(x\right)+x^{13}+10x^{12}-50x^{11}+23x^{10}+200x^9-386x^8+119x^7+464x^6-774x^5+614x^4-292x^3+85x^2-14x+1=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=21}$
${\displaystyle a\left(5\right)=76}$
${\displaystyle a\left(6\right)=275}$
${\displaystyle a\left(7\right)=989}$
${\displaystyle a\left(8\right)=3544}$
${\displaystyle a\left(9\right)=12696}$
${\displaystyle a\left(10\right)=45578}$
${\displaystyle a\left(11\right)=164194}$
${\displaystyle a\left(12\right)=593966}$
${\displaystyle a\left(13\right)=2158090}$
${\displaystyle a(n+13)=\frac{2(1+2n)a\left(n\right)}{15+n}+\frac{(217+80n)a(2+n)}{15+n}-\frac{3(18+11n)a(n+1)}{15+n}+\frac{(155+34n)a(n+3)}{15+n}-\frac{(2277+445n)a(n+4)}{15+n}+\frac{2(2071+324n)a(n+5)}{15+n}-\frac{(794+27n)a(n+6)}{15+n}-\frac{(7053+959n)a(n+7)}{15+n}+\frac{(11816+1333n)a(n+8)}{15+n}-\frac{(9713+955n)a(n+9)}{15+n}+\frac{2(2363+207n)a(n+10)}{15+n}-\frac{(1376+109n)a(n+11)}{15+n}+\frac{(221+16n)a(n+12)}{15+n}+\frac{4}{15+n},n\ge 14}$