Av(1243, 1324, 1342, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(x -1\right)^{5} \sqrt{-4 x +1}-2 x^{6}-5 x^{5}+11 x^{4}-12 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 178, 561, 1821, 6064, 20616, 71268, 249745, 885117, 3166842, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(x^{5}+3 x^{4}-4 x^{3}+4 x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{11}+6 x^{10}-15 x^{9}+32 x^{8}-81 x^{7}+154 x^{6}-202 x^{5}+183 x^{4}-112 x^{3}+44 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 178\)
\(\displaystyle a \! \left(7\right) = 561\)
\(\displaystyle a \! \left(n +3\right) = \frac{4 \left(2 n -1\right) a \! \left(n \right)}{4+n}-\frac{2 \left(7 n +10\right) a \! \left(1+n \right)}{4+n}+\frac{\left(7 n +19\right) a \! \left(n +2\right)}{4+n}+\frac{15 n^{2}-77 n +64}{2 n +8}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 178\)
\(\displaystyle a \! \left(7\right) = 561\)
\(\displaystyle a \! \left(n +3\right) = \frac{4 \left(2 n -1\right) a \! \left(n \right)}{4+n}-\frac{2 \left(7 n +10\right) a \! \left(1+n \right)}{4+n}+\frac{\left(7 n +19\right) a \! \left(n +2\right)}{4+n}+\frac{15 n^{2}-77 n +64}{2 n +8}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 34 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 34 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{5} \left(x \right)^{3}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{26}\! \left(x \right) &= 0\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{23}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)