Av(1342, 2314, 3412)
Generating Function
\(\displaystyle \frac{-9 \left(x^{2}-x +\frac{1}{3}\right)^{2} \sqrt{1-4 x}-8 x^{6}+18 x^{5}-7 x^{4}-10 x^{3}+13 x^{2}-6 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 262, 894, 3011, 10120, 34213, 116864, 404013, 1413582, 5000943, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{10} F \left(x
\right)^{2}+\left(4 x^{5}-7 x^{4}+5 x^{2}-4 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+16 x^{11}-72 x^{10}+109 x^{9}+58 x^{8}-474 x^{7}+851 x^{6}-875 x^{5}+586 x^{4}-262 x^{3}+76 x^{2}-13 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 262\)
\(\displaystyle a \! \left(7\right) = 894\)
\(\displaystyle a \! \left(8\right) = 3011\)
\(\displaystyle a \! \left(n +5\right) = \frac{12 \left(2 n +5\right) a \! \left(n \right)}{n +6}-\frac{6 \left(11 n +25\right) a \! \left(n +1\right)}{n +6}+\frac{\left(206+71 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(38 n +147\right) a \! \left(n +3\right)}{n +6}+\frac{\left(49+10 n \right) a \! \left(n +4\right)}{n +6}+\frac{2 n^{3}-24 n^{2}+19 n -57}{3 n +18}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 262\)
\(\displaystyle a \! \left(7\right) = 894\)
\(\displaystyle a \! \left(8\right) = 3011\)
\(\displaystyle a \! \left(n +5\right) = \frac{12 \left(2 n +5\right) a \! \left(n \right)}{n +6}-\frac{6 \left(11 n +25\right) a \! \left(n +1\right)}{n +6}+\frac{\left(206+71 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(38 n +147\right) a \! \left(n +3\right)}{n +6}+\frac{\left(49+10 n \right) a \! \left(n +4\right)}{n +6}+\frac{2 n^{3}-24 n^{2}+19 n -57}{3 n +18}, \quad n \geq 9\)
This specification was found using the strategy pack "Insertion Point Placements" and has 63 rules.
Found on July 23, 2021.Finding the specification took 12 seconds.
Copy 63 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_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{6}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{20}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{20}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{20}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{20}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 0\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{20} \left(x \right)^{2} F_{7}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{28}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{20} \left(x \right)^{2} F_{22}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{20}\! \left(x \right) F_{28}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{20} \left(x \right)^{2} F_{37}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{58}\! \left(x \right)\\
\end{align*}\)