###### Av(12435, 12453, 12543, 14235, 14253, 14523, 15243, 15423, 21435, 21453, 21543, 25143, 51243, 51423, 52143)
Generating Function
$$\displaystyle \frac{\left(-7 x^{2}+5 x \right) \sqrt{5 x^{2}-6 x +1}+10 x^{4}-37 x^{3}+72 x^{2}-53 x +8}{10 x^{4}-52 x^{3}+90 x^{2}-56 x +8}$$
Counting Sequence
1, 1, 2, 6, 24, 105, 474, 2177, 10120, 47475, 224310, 1065900, 5088816, 24389859, 117282186, ...
Implicit Equation for the Generating Function
$$\displaystyle \left(5 x -1\right) \left(x -1\right) \left(x -2\right)^{2} F \left(x \right)^{2}-\left(5 x -1\right) \left(x -1\right) \left(2 x^{2}-5 x +8\right) F \! \left(x \right)+5 x^{4}-11 x^{3}+26 x^{2}-25 x +4 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(2\right) = 2$$
$$\displaystyle a \! \left(3\right) = 6$$
$$\displaystyle a \! \left(4\right) = 24$$
$$\displaystyle a \! \left(n +4\right) = -\frac{7 \left(n +1\right) a \! \left(n \right)}{2 \left(3+n \right)}+\frac{\left(184+79 n \right) a \! \left(3+n \right)}{30+10 n}+\frac{\left(138+137 n \right) a \! \left(n +1\right)}{30+10 n}-\frac{9 \left(29+19 n \right) a \! \left(n +2\right)}{10 \left(3+n \right)}, \quad n \geq 5$$

### This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 23 rules.

Found on January 23, 2022.

Finding the specification took 0 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{22}\! \left(x \right) &= x\\ \end{align*}