Av(1243, 1324, 1342, 1423, 2134, 2341, 4123)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(-x^{2}-x +1\right) \sqrt{1-4 x}-2 x^{6}-4 x^{5}-2 x^{4}+x^{2}+x -1}{2 \left(x^{2}+x -1\right) x}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 17, 47, 140, 442, 1451, 4896, 16851, 58875, 208156, 743133, 2674817, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}+x -1\right)^{2} F \left(x \right)^{2}+\left(x^{2}+x -1\right) \left(2 x^{6}+4 x^{5}+2 x^{4}-x^{2}-x +1\right) F \! \left(x \right)+x^{11}+4 x^{10}+6 x^{9}+4 x^{8}-3 x^{6}-2 x^{5}+2 x^{4}+3 x^{3}-x^{2}-2 x +1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 47\)
\(\displaystyle a \! \left(6\right) = 140\)
\(\displaystyle a \! \left(7\right) = 442\)
\(\displaystyle a \! \left(8\right) = 1451\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +6}+\frac{\left(10+7 n \right) a \! \left(1+n \right)}{n +6}-\frac{2 \left(8+3 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(24+7 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(14+3 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 9\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 42 rules.

Found on July 23, 2021.

Finding the specification took 4 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

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_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{5}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \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_{14}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x , 1\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{38}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x , y\right)\\ F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{31}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= y x\\ F_{38}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{41}\! \left(x , y\right) &= \frac{y F_{27}\! \left(x , y\right)-F_{27}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)