Permutations avoiding 132, 643257819

Proof tree for permutations avoiding 132, 643257819

Legend

A = Av(120, 865041237)

B = Av(120, 65041237)

Equations:
${ F_{0}}={ F_{1}}+{ F_{3}}$
${ F_{3}}={ F_{0}}{ F_{12}}{ F_{8}}$
${ F_{12}}={\frac {{x}^{5}-10{x}^{4}+19{x}^{3}-17{x}^{2}+7x-1}{5{x}^{5}-21{x}^{4}+31{x}^{3}-23{x}^{2}+8x-1}}$
${ F_{8}}=x$
${ F_{1}}=1$

Coefficients:
$1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861,\ldots$

Minimal polynomial:
$ \left( x-1 \right) \left( {x}^{5}-14{x}^{4}+26{x}^{3}-22{x}^{2}+8x-1 \right) F \left( x \right) +5{x}^{5}-21{x}^{4}+31{x}^{3}-23{x}^{2}+8x-1$

Generating function:
$-{\frac {5{x}^{5}-21{x}^{4}+31{x}^{3}-23{x}^{2}+8x-1}{ \left( x-1 \right) \left( {x}^{5}-14{x}^{4}+26{x}^{3}-22{x}^{2}+8x-1 \right) }}$

Recurrence relation:
$-a \left( n \right) +14a \left( n+1 \right) -26a \left( n+2 \right) +22a \left( n+3 \right) -8a \left( n+4 \right) +a \left( n+5 \right) -1$
$a \left( 0 \right) =1$
$a \left( 1 \right) =1$
$a \left( 2 \right) =2$
$a \left( 3 \right) =5$
$a \left( 4 \right) =14$
$a \left( 5 \right) =42$

Closed form:
N/A