Av(1234, 13524, 14523, 23514, 24513, 34512)
Counting Sequence
1, 1, 2, 6, 23, 98, 434, 1949, 8803, 39888, 181201, 825201, 3767757, 17249560, 79191480, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} x^{5} F \left(x
\right)^{4}-2 x^{3} \left(4 x -1\right) \left(x -1\right)^{2} F \left(x
\right)^{3}+x \left(x -1\right) \left(2 x^{4}+15 x^{3}-28 x^{2}+10 x -1\right) F \left(x
\right)^{2}-\left(x -1\right) \left(4 x -1\right) \left(2 x^{3}-x^{2}-4 x +1\right) F \! \left(x \right)+x^{5}+4 x^{4}-21 x^{3}+25 x^{2}-9 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 23\)
\(\displaystyle a \! \left(5\right) = 98\)
\(\displaystyle a \! \left(6\right) = 434\)
\(\displaystyle a \! \left(n +7\right) = -\frac{100 \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{\left(n +9\right) \left(n +8\right)}+\frac{10 \left(17 n +30\right) \left(n -1\right) a \! \left(n +1\right)}{\left(n +9\right) \left(n +8\right)}+\frac{2 \left(287 n^{2}+2157 n +3880\right) a \! \left(n +2\right)}{\left(n +9\right) \left(n +8\right)}-\frac{\left(975 n^{2}+8471 n +18196\right) a \! \left(n +3\right)}{\left(n +9\right) \left(n +8\right)}+\frac{2 \left(283 n^{2}+2999 n +7860\right) a \! \left(n +4\right)}{\left(n +9\right) \left(n +8\right)}-\frac{2 \left(77 n^{2}+975 n +3058\right) a \! \left(n +5\right)}{\left(n +9\right) \left(n +8\right)}+\frac{4 \left(5 n +34\right) a \! \left(n +6\right)}{n +9}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 23\)
\(\displaystyle a \! \left(5\right) = 98\)
\(\displaystyle a \! \left(6\right) = 434\)
\(\displaystyle a \! \left(n +7\right) = -\frac{100 \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{\left(n +9\right) \left(n +8\right)}+\frac{10 \left(17 n +30\right) \left(n -1\right) a \! \left(n +1\right)}{\left(n +9\right) \left(n +8\right)}+\frac{2 \left(287 n^{2}+2157 n +3880\right) a \! \left(n +2\right)}{\left(n +9\right) \left(n +8\right)}-\frac{\left(975 n^{2}+8471 n +18196\right) a \! \left(n +3\right)}{\left(n +9\right) \left(n +8\right)}+\frac{2 \left(283 n^{2}+2999 n +7860\right) a \! \left(n +4\right)}{\left(n +9\right) \left(n +8\right)}-\frac{2 \left(77 n^{2}+975 n +3058\right) a \! \left(n +5\right)}{\left(n +9\right) \left(n +8\right)}+\frac{4 \left(5 n +34\right) a \! \left(n +6\right)}{n +9}, \quad n \geq 7\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Req Corrob Expand Verified" and has 33 rules.
Found on January 22, 2022.Finding the specification took 26 seconds.
Copy 33 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{5}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\
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_{13}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{5}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{18}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{22}\! \left(x , y\right) &= -\frac{-y F_{20}\! \left(x , y\right)+F_{20}\! \left(x , 1\right)}{-1+y}\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27} \left(x \right)^{2} F_{3}\! \left(x \right) F_{30}\! \left(x , y\right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Expand Verified" and has 35 rules.
Found on January 22, 2022.Finding the specification took 27 seconds.
Copy 35 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 , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{15}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\
F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= -\frac{-y F_{22}\! \left(x , y\right)+F_{22}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29} \left(x \right)^{2} F_{32}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\
\end{align*}\)