Av(1243, 1342, 2134, 3124)
View Raw Data
Generating Function
\(\displaystyle \frac{-\left(x^{3}+x^{2}-x +1\right)^{2} \sqrt{1-4 x}+3 x^{6}+6 x^{5}-x^{4}+2 x^{3}+5 x^{2}-4 x +1}{2 x^{7}+8 x^{6}+6 x^{5}+8 x^{3}}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 20, 66, 220, 749, 2596, 9116, 32346, 115795, 417734, 1517112, 5542232, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(x +2\right) \left(x^{3}+2 x^{2}-x +2\right) F \left(x \right)^{2}+\left(-3 x^{6}-6 x^{5}+x^{4}-2 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{6}+2 x^{5}-x^{4}+x^{3}+3 x^{2}-3 x +1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 220\)
\(\displaystyle a \! \left(7\right) = 749\)
\(\displaystyle a \! \left(n +8\right) = \frac{\left(1+2 n \right) a \! \left(n \right)}{22+2 n}-\frac{3 \left(n +7\right) a \! \left(3+n \right)}{2 \left(11+n \right)}+\frac{\left(20+19 n \right) a \! \left(n +1\right)}{44+4 n}+\frac{\left(23+19 n \right) a \! \left(n +2\right)}{44+4 n}+\frac{\left(22+5 n \right) a \! \left(n +4\right)}{11+n}+\frac{\left(142+23 n \right) a \! \left(n +5\right)}{44+4 n}-\frac{\left(157+23 n \right) a \! \left(n +6\right)}{4 \left(11+n \right)}+\frac{\left(46+5 n \right) a \! \left(n +7\right)}{11+n}, \quad n \geq 8\)
Copy to clipboard:

This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 34 rules.

Found on July 23, 2021.

Finding the specification took 8 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 34 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_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{27}\! \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_{9}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \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_{16}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{20}\! \left(x , y\right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{5}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= y x\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{26}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= \frac{y F_{18}\! \left(x , y\right)-F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ \end{align*}\)