Permutations avoiding 132, 2134, 2314, 3241, 3412, 4213, 4231

Proof tree for permutations avoiding 132, 2134, 2314, 3241, 3412, 4213, 4231

Legend

A = Av(021, 1023, 1203, 2130, 2301, 3102, 3120)

B = Av(021, 102, 201)

C = Av(021, 102, 120)

Equations:
${ F_{0}}={ F_{1}}+{ F_{3}}$
${ F_{1}}=1$
${ F_{3}}={\frac {2{x}^{3}+x}{ \left( x-1 \right) ^{2}}}$

Coefficients:
$1, 1, 2, 5, 8, 11, 14, 17, 20, 23,\ldots$

Minimal polynomial:
$ \left( x-1 \right) ^{2}F \left( x \right) -2{x}^{3}-{x}^{2}+x-1$

Generating function:
${\frac {2{x}^{3}+{x}^{2}-x+1}{ \left( x-1 \right) ^{2}}}$

Recurrence relation:
$a \left( n \right) -3n+4$
$a \left( 0 \right) =1$
$a \left( 1 \right) =1$
$a \left( 2 \right) =2$
$a \left( 3 \right) =5$

Closed form:
$\cases{1&$n=0$\cr 1&$n=1$\cr -4+3n&otherwise\cr}$