${\displaystyle \frac{(2x^4-9x^3+16x^2-10x+2)\sqrt{5x^2-6x+1}-4x^5+15x^4-17x^3-2x^2+8x-2}{4x{(x-2)}^2(x-\frac{1}{2})(x^2-3x+1)}}$

1, 1, 2, 6, 22, 88, 365, 1540, 6568, 28269, 122752, 537708, 2375500, 10579400, 47469377, ...

${\displaystyle x{(x-2)}^2{(2x-1)}^2{(x^2-3x+1)}^{2}F{\left(x\right)}^2+(2x-1)(x^2-3x+1)(4x^5-15x^4+17x^3+2x^2-8x+2)F\left(x\right)-x^7+17x^6-74x^5+149x^4-150x^3+78x^2-20x+2=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=22}$
${\displaystyle a\left(5\right)=88}$
${\displaystyle a\left(6\right)=365}$
${\displaystyle a\left(7\right)=1540}$
${\displaystyle a\left(8\right)=6568}$
${\displaystyle a\left(9\right)=28269}$
${\displaystyle a(n+10)=-\frac{5(1+n)a\left(n\right)}{11+n}+\frac{4(25+14n)a(1+n)}{11+n}-\frac{(2939+1089n)a(n+2)}{4(11+n)}+\frac{(11020+2999n)a(n+3)}{44+4n}-\frac{(11927+2554n)a(n+4)}{2(11+n)}+\frac{5(6299+1109n)a(n+5)}{4(11+n)}-\frac{(25862+3857n)a(n+6)}{4(11+n)}+\frac{4(821+106n)a(n+7)}{11+n}-\frac{(1989+226n)a(n+8)}{2(11+n)}+\frac{(326+33n)a(n+9)}{22+2n},n\ge 10}$

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point $(i,j)$ represents how many permutations have value $j$ at index $i$ (darker = more).