Av(12354, 12453, 12543, 21354, 21453, 21543, 31254, 31452, 31542, 41253, 41352, 41532, 51243, 51342, 51432)
Generating Function
\(\displaystyle \frac{\left(-7 x^{2}+5 x \right) \sqrt{5 x^{2}-6 x +1}+10 x^{4}-37 x^{3}+72 x^{2}-53 x +8}{10 x^{4}-52 x^{3}+90 x^{2}-56 x +8}\)
Counting Sequence
1, 1, 2, 6, 24, 105, 474, 2177, 10120, 47475, 224310, 1065900, 5088816, 24389859, 117282186, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(5 x -1\right) \left(x -1\right) \left(x -2\right)^{2} F \left(x
\right)^{2}-\left(5 x -1\right) \left(x -1\right) \left(2 x^{2}-5 x +8\right) F \! \left(x \right)+5 x^{4}-11 x^{3}+26 x^{2}-25 x +4 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +4\right) = -\frac{7 \left(n +1\right) a \! \left(n \right)}{2 \left(3+n \right)}+\frac{\left(184+79 n \right) a \! \left(3+n \right)}{30+10 n}+\frac{\left(138+137 n \right) a \! \left(n +1\right)}{30+10 n}-\frac{9 \left(29+19 n \right) a \! \left(n +2\right)}{10 \left(3+n \right)}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(n +4\right) = -\frac{7 \left(n +1\right) a \! \left(n \right)}{2 \left(3+n \right)}+\frac{\left(184+79 n \right) a \! \left(3+n \right)}{30+10 n}+\frac{\left(138+137 n \right) a \! \left(n +1\right)}{30+10 n}-\frac{9 \left(29+19 n \right) a \! \left(n +2\right)}{10 \left(3+n \right)}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 23 rules.
Found on January 23, 2022.Finding the specification took 1 seconds.
Copy 23 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= \frac{y F_{7}\! \left(x , y\right)-F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= y x\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
\end{align*}\)