Av(12435, 12453, 12543, 14235, 14253, 41235, 41253)
Counting Sequence
1, 1, 2, 6, 24, 113, 580, 3129, 17442, 99574, 579108, 3419056, 20440024, 123494294, 752913720, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x +2\right) \left(x -1\right)^{4} F \left(x
\right)^{4}+\left(2 x^{2}+10 x -9\right) \left(x -1\right)^{2} F \left(x
\right)^{3}+\left(2 x^{4}-7 x^{3}+31 x^{2}-40 x +15\right) F \left(x
\right)^{2}+\left(2 x^{3}-14 x^{2}+24 x -11\right) F \! \left(x \right)+x^{3}+2 x^{2}-5 x +3 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 113\)
\(\displaystyle a \! \left(6\right) = 580\)
\(\displaystyle a \! \left(7\right) = 3129\)
\(\displaystyle a \! \left(8\right) = 17442\)
\(\displaystyle a \! \left(9\right) = 99574\)
\(\displaystyle a \! \left(10\right) = 579108\)
\(\displaystyle a \! \left(11\right) = 3419056\)
\(\displaystyle a \! \left(n +12\right) = -\frac{2048 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{6144 \left(3 n +10\right) \left(2 n +3\right) \left(n +2\right) a \! \left(n +1\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{768 \left(34 n^{3}+588 n^{2}+2185 n +2396\right) a \! \left(n +2\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{384 \left(248 n^{3}+986 n^{2}-526 n -4023\right) a \! \left(n +3\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{96 \left(2254 n^{3}+20052 n^{2}+57283 n +51141\right) a \! \left(n +4\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{96 \left(2039 n^{3}+24143 n^{2}+94272 n +121515\right) a \! \left(n +5\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{4 \left(23030 n^{3}+331212 n^{2}+1566151 n +2435784\right) a \! \left(n +6\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{6 \left(3572 n^{3}+57954 n^{2}+303012 n +504617\right) a \! \left(n +7\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{3 \left(199 n^{3}-368 n^{2}-38281 n -181311\right) a \! \left(n +8\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(1937 n^{3}+52080 n^{2}+464680 n +1375884\right) a \! \left(n +9\right)}{2 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{3 \left(164 n^{2}+2991 n +13631\right) a \! \left(n +10\right)}{2 \left(n +12\right) \left(n +11\right)}+\frac{3 \left(17 n +162\right) a \! \left(n +11\right)}{2 \left(n +12\right)}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 113\)
\(\displaystyle a \! \left(6\right) = 580\)
\(\displaystyle a \! \left(7\right) = 3129\)
\(\displaystyle a \! \left(8\right) = 17442\)
\(\displaystyle a \! \left(9\right) = 99574\)
\(\displaystyle a \! \left(10\right) = 579108\)
\(\displaystyle a \! \left(11\right) = 3419056\)
\(\displaystyle a \! \left(n +12\right) = -\frac{2048 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{6144 \left(3 n +10\right) \left(2 n +3\right) \left(n +2\right) a \! \left(n +1\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{768 \left(34 n^{3}+588 n^{2}+2185 n +2396\right) a \! \left(n +2\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{384 \left(248 n^{3}+986 n^{2}-526 n -4023\right) a \! \left(n +3\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{96 \left(2254 n^{3}+20052 n^{2}+57283 n +51141\right) a \! \left(n +4\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{96 \left(2039 n^{3}+24143 n^{2}+94272 n +121515\right) a \! \left(n +5\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{4 \left(23030 n^{3}+331212 n^{2}+1566151 n +2435784\right) a \! \left(n +6\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{6 \left(3572 n^{3}+57954 n^{2}+303012 n +504617\right) a \! \left(n +7\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{3 \left(199 n^{3}-368 n^{2}-38281 n -181311\right) a \! \left(n +8\right)}{\left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(1937 n^{3}+52080 n^{2}+464680 n +1375884\right) a \! \left(n +9\right)}{2 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{3 \left(164 n^{2}+2991 n +13631\right) a \! \left(n +10\right)}{2 \left(n +12\right) \left(n +11\right)}+\frac{3 \left(17 n +162\right) a \! \left(n +11\right)}{2 \left(n +12\right)}, \quad n \geq 12\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 45 rules.
Found on January 23, 2022.Finding the specification took 226 seconds.
Copy 45 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_{9}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\
F_{4}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x , y\right)\\
F_{5}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{4}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{38}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{4}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y\right)+F_{26}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{25}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\
F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= -\frac{-y F_{4}\! \left(x , y\right)+F_{4}\! \left(x , 1\right)}{-1+y}\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{29}\! \left(x , y\right) &= -\frac{-y F_{27}\! \left(x , y\right)+F_{27}\! \left(x , 1\right)}{-1+y}\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{31}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x \right) F_{4}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38} \left(x \right)^{2} F_{9}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{38}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{38}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{42}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 30 rules.
Found on January 22, 2022.Finding the specification took 10 seconds.
Copy 30 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_{20}\! \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_{21}\! \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 , z\right)\\
F_{15}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y , z\right)+F_{16}\! \left(x , y , z\right)+F_{18}\! \left(x , y , z\right)\\
F_{15}\! \left(x , y , z\right) &= \frac{y F_{4}\! \left(x , y\right)-z F_{4}\! \left(x , z\right)}{-z +y}\\
F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right) F_{7}\! \left(x \right)\\
F_{17}\! \left(x , y , z\right) &= -\frac{-y F_{15}\! \left(x , y , z\right)+F_{15}\! \left(x , 1, z\right)}{-1+y}\\
F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right) F_{20}\! \left(x , z\right)\\
F_{19}\! \left(x , y , z\right) &= -\frac{z F_{10}\! \left(x , 1, z\right)-y F_{10}\! \left(x , \frac{y}{z}, z\right)}{-z +y}\\
F_{20}\! \left(x , y\right) &= y x\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{20}\! \left(x , z\right) F_{23}\! \left(x , z\right) F_{4}\! \left(x , z\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)^{2} F_{20}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
\end{align*}\)