Av(12453, 12543, 14253, 14523, 15243, 15423, 41253, 41523, 45123, 51243, 51423, 54123)
Generating Function
\(\displaystyle \frac{2 x^{3}-5 x^{2}-\sqrt{12 x^{2}-8 x +1}+5 x +1}{x \left(2 x -3\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 108, 512, 2506, 12560, 64148, 332704, 1747748, 9280416, 49731768, 268613568, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -3\right)^{2} F \left(x
\right)^{2}+\left(-4 x^{3}+10 x^{2}-10 x -2\right) F \! \left(x \right)+x^{3}-2 x^{2}+3 x +2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(n +2\right) a \! \left(n \right)}{n +4}-\frac{4 \left(13 n +23\right) a \! \left(1+n \right)}{3 \left(n +4\right)}+\frac{2 \left(13 n +35\right) a \! \left(n +2\right)}{3 \left(n +4\right)}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(n +2\right) a \! \left(n \right)}{n +4}-\frac{4 \left(13 n +23\right) a \! \left(1+n \right)}{3 \left(n +4\right)}+\frac{2 \left(13 n +35\right) a \! \left(n +2\right)}{3 \left(n +4\right)}, \quad n \geq 4\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 61 rules.
Found on January 23, 2022.Finding the specification took 12 seconds.
Copy 61 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_{17}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)+F_{40}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= y x\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x , y\right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{31}\! \left(x \right) &= 0\\
F_{32}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{36}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{37}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{40}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x , y\right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{17}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{47}\! \left(x , y\right) &= 2 F_{31}\! \left(x \right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\
F_{51}\! \left(x \right) &= F_{17}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\
F_{53}\! \left(x \right) &= F_{17}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{17}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x , 1\right)\\
F_{57}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x , 1\right)\\
F_{59}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{17}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 42 rules.
Found on January 22, 2022.Finding the specification took 8 seconds.
Copy 42 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_{7}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{6}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x \right) &= x\\
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 , 1, y\right)\\
F_{10}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y , z\right)+F_{13}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)\\
F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right) F_{7}\! \left(x \right)\\
F_{12}\! \left(x , y , z\right) &= \frac{y z F_{10}\! \left(x , y , z\right)-F_{10}\! \left(x , \frac{1}{z}, z\right)}{y z -1}\\
F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , z\right) F_{17}\! \left(x , z\right)\\
F_{14}\! \left(x , y , z\right) &= \frac{y F_{15}\! \left(x , y z , 1\right)-F_{15}\! \left(x , y z , \frac{1}{y}\right)}{-1+y}\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , y z \right)\\
F_{10}\! \left(x , y , z\right) &= F_{16}\! \left(x , y z , z\right)\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{19}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , z\right)+F_{36}\! \left(x , y , z\right)\\
F_{20}\! \left(x , y , z\right) &= F_{21}\! \left(x , z\right)+F_{31}\! \left(x , y , z\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{27}\! \left(x \right) &= 0\\
F_{28}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{31}\! \left(x , y , z\right) &= F_{32}\! \left(x , y , z\right)\\
F_{32}\! \left(x , y , z\right) &= F_{17}\! \left(x , y z \right)+F_{33}\! \left(x , y , z\right)\\
F_{33}\! \left(x , y , z\right) &= F_{27}\! \left(x \right)+F_{34}\! \left(x , y , z\right)+F_{35}\! \left(x , y , z\right)\\
F_{34}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{23}\! \left(x , z\right)\\
F_{35}\! \left(x , y , z\right) &= F_{17}\! \left(x , z\right) F_{31}\! \left(x , y , z\right)\\
F_{36}\! \left(x , y , z\right) &= F_{37}\! \left(x , z\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{40}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{41}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\
\end{align*}\)