Av(1342, 3142, 45132)
View Raw Data
Counting Sequence
1, 1, 2, 6, 22, 89, 380, 1678, 7584, 34875, 162560, 766124, 3644066, 17469863, 84324840, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}-2 x +2\right) F \left(x \right)^{4}+\left(2 x^{2}-4 x -1\right) F \left(x \right)^{3}+\left(2 x +3\right) F \left(x \right)^{2}-3 F \! \left(x \right)+1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 89\)
\(\displaystyle a \! \left(6\right) = 380\)
\(\displaystyle a \! \left(7\right) = 1678\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(4 n +5\right) \left(2 n +3\right) \left(4 n +3\right) a \! \left(n \right)}{9 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(14276 n^{3}+105300 n^{2}+252862 n +196515\right) a \! \left(n +1\right)}{27 \left(n +9\right) \left(n +7\right) \left(n +6\right)}-\frac{\left(88628 n^{3}+756114 n^{2}+2152783 n +2046777\right) a \! \left(n +2\right)}{54 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(271825 n^{3}+2607174 n^{2}+8461187 n +9320358\right) a \! \left(n +3\right)}{108 \left(n +9\right) \left(n +7\right) \left(n +6\right)}-\frac{\left(200183 n^{3}+2331978 n^{2}+9033313 n +11721078\right) a \! \left(n +4\right)}{108 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(20005 n^{3}+283212 n^{2}+1323137 n +2050536\right) a \! \left(n +5\right)}{27 \left(n +9\right) \left(n +7\right) \left(n +6\right)}-\frac{2 \left(2258 n^{3}+37968 n^{2}+210241 n +384615\right) a \! \left(n +6\right)}{27 \left(n +9\right) \left(n +7\right) \left(n +6\right)}+\frac{\left(181 n^{2}+2436 n +7931\right) a \! \left(n +7\right)}{9 \left(n +7\right) \left(n +9\right)}, \quad n \geq 8\)
Copy to clipboard:

This specification was found using the strategy pack "Row And Col Placements Expand Verified" and has 13 rules.

Found on January 22, 2022.

Finding the specification took 6 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 13 equations to clipboard:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{3}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{3}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "All The Strategies 1 Expand Verified" and has 14 rules.

Found on January 22, 2022.

Finding the specification took 24 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 14 equations to clipboard:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{4}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row Placements Req Corrob Expand Verified" and has 14 rules.

Found on January 22, 2022.

Finding the specification took 9 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 14 equations to clipboard:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{7}\! \left(x \right)\\ \end{align*}\)