ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(132, 1234, 2134, 2314, 3214, 3241, 4213)

$\mathcal{B}$ = Av(123, 132, 213, 231, 321)

$\mathcal{C}$ = Av(132, 213, 1234)

$\mathcal{D}$ = Av(21, 123)

Generating function

$A(x) = \frac{x^{7} + 2 x^{6} + 2 x^{5} - x^{3} - 1}{x^{3} + x^{2} + x - 1}$

Coefficients

1, 1, 2, 5, 8, 13, 24, 44, 81, 149, 274, ...

System of equations

$\operatorname{F_{4}}{\left (x \right )} = \operatorname{F_{0}}{\left (x \right )} + \operatorname{F_{3}}{\left (x \right )}$

$\operatorname{F_{0}}{\left (x \right )} = 1$

$\operatorname{F_{3}}{\left (x \right )} = \operatorname{F_{1}}{\left (x \right )} + \operatorname{F_{2}}{\left (x \right )}$

$\operatorname{F_{1}}{\left (x \right )} = x \left(x^{3} + 2 x^{2} + x + 1\right)$

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