Av(1342, 2143, 2431, 3142)
Generating Function
\(\displaystyle \frac{-2 \left(x -1\right)^{5} \left(x -\frac{1}{2}\right) \sqrt{1-4 x}-4 x^{6}+3 x^{5}+7 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{8 x^{7}-42 x^{6}+82 x^{5}-88 x^{4}+52 x^{3}-16 x^{2}+2 x}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 224, 752, 2544, 8683, 29909, 103965, 364595, 1289413, 4596270, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) \left(x^{2}-3 x +1\right)^{2} F \left(x
\right)^{2}+\left(x^{2}-3 x +1\right) \left(4 x^{6}-3 x^{5}-7 x^{4}+20 x^{3}-18 x^{2}+7 x -1\right) F \! \left(x \right)+x^{8}-8 x^{7}+36 x^{6}-82 x^{5}+111 x^{4}-89 x^{3}+41 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 224\)
\(\displaystyle a \! \left(7\right) = 752\)
\(\displaystyle a \! \left(8\right) = 2544\)
\(\displaystyle a \! \left(n +9\right) = \frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +10}-\frac{4 \left(56 n +87\right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(325 n +883\right) a \! \left(n +2\right)}{n +10}-\frac{3 \left(359 n +1352\right) a \! \left(n +3\right)}{n +10}+\frac{4 \left(283 n +1358\right) a \! \left(n +4\right)}{n +10}-\frac{\left(777 n +4534\right) a \! \left(n +5\right)}{n +10}+\frac{2 \left(173 n +1188\right) a \! \left(n +6\right)}{n +10}-\frac{2 \left(48 n +379\right) a \! \left(n +7\right)}{n +10}+\frac{\left(134+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 224\)
\(\displaystyle a \! \left(7\right) = 752\)
\(\displaystyle a \! \left(8\right) = 2544\)
\(\displaystyle a \! \left(n +9\right) = \frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +10}-\frac{4 \left(56 n +87\right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(325 n +883\right) a \! \left(n +2\right)}{n +10}-\frac{3 \left(359 n +1352\right) a \! \left(n +3\right)}{n +10}+\frac{4 \left(283 n +1358\right) a \! \left(n +4\right)}{n +10}-\frac{\left(777 n +4534\right) a \! \left(n +5\right)}{n +10}+\frac{2 \left(173 n +1188\right) a \! \left(n +6\right)}{n +10}-\frac{2 \left(48 n +379\right) a \! \left(n +7\right)}{n +10}+\frac{\left(134+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 37 rules.
Found on July 23, 2021.Finding the specification took 7 seconds.
Copy 37 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{0}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= 0\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{2}\! \left(x \right)\\
\end{align*}\)