Av(1342, 2314, 2431)
Generating Function
\(\displaystyle \frac{\left(2 x^{4}-5 x^{3}+8 x^{2}-5 x +1\right) \sqrt{1-4 x}+2 x^{5}-10 x^{4}+21 x^{3}-18 x^{2}+7 x -1}{2 x^{2} \left(x^{2}-3 x +1\right) \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 267, 948, 3367, 11988, 42842, 153783, 554624, 2009904, 7318260, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x^{5}-10 x^{4}+21 x^{3}-18 x^{2}+7 x -1\right) F \! \left(x \right)+x^{8}-6 x^{7}+25 x^{6}-61 x^{5}+93 x^{4}-82 x^{3}+40 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 267\)
\(\displaystyle a \! \left(7\right) = 948\)
\(\displaystyle a \! \left(8\right) = 3367\)
\(\displaystyle a \! \left(n +8\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{10+n}-\frac{\left(396+157 n \right) a \! \left(2+n \right)}{10+n}+\frac{2 \left(35+27 n \right) a \! \left(n +1\right)}{10+n}+\frac{34 \left(31+8 n \right) a \! \left(n +3\right)}{10+n}-\frac{\left(1496+291 n \right) a \! \left(n +4\right)}{10+n}+\frac{6 \left(197+31 n \right) a \! \left(n +5\right)}{10+n}-\frac{2 \left(257+34 n \right) a \! \left(n +6\right)}{10+n}+\frac{\left(114+13 n \right) a \! \left(n +7\right)}{10+n}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 267\)
\(\displaystyle a \! \left(7\right) = 948\)
\(\displaystyle a \! \left(8\right) = 3367\)
\(\displaystyle a \! \left(n +8\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{10+n}-\frac{\left(396+157 n \right) a \! \left(2+n \right)}{10+n}+\frac{2 \left(35+27 n \right) a \! \left(n +1\right)}{10+n}+\frac{34 \left(31+8 n \right) a \! \left(n +3\right)}{10+n}-\frac{\left(1496+291 n \right) a \! \left(n +4\right)}{10+n}+\frac{6 \left(197+31 n \right) a \! \left(n +5\right)}{10+n}-\frac{2 \left(257+34 n \right) a \! \left(n +6\right)}{10+n}+\frac{\left(114+13 n \right) a \! \left(n +7\right)}{10+n}, \quad n \geq 9\)
This specification was found using the strategy pack "Row Placements" and has 43 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 43 equations to clipboard:
\(\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_{14}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{4}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{14}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{25}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
\end{align*}\)