Av(1243, 1342, 2413, 3142)
Generating Function
\(\displaystyle \frac{-x^{3}+2 x^{2}-3 x +1-\sqrt{x^{6}-20 x^{5}+58 x^{4}-66 x^{3}+37 x^{2}-10 x +1}}{4 x^{3}-6 x^{2}+2 x}\)
Counting Sequence
1, 1, 2, 6, 20, 68, 234, 818, 2910, 10528, 38666, 143868, 541312, 2056354, 7876802, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right) \left(x -1\right) F \left(x
\right)^{2}+\left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 234\)
\(\displaystyle a \! \left(7\right) = 818\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 n a \! \left(n \right)}{9+n}+\frac{\left(43 n +60\right) a \! \left(1+n \right)}{9+n}-\frac{3 \left(59 n +146\right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(163 n +573\right) a \! \left(n +3\right)}{9+n}-\frac{3 \left(110 n +503\right) a \! \left(n +4\right)}{9+n}+\frac{\left(197 n +1113\right) a \! \left(n +5\right)}{9+n}-\frac{3 \left(23 n +155\right) a \! \left(n +6\right)}{9+n}+\frac{\left(13 n +102\right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 234\)
\(\displaystyle a \! \left(7\right) = 818\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 n a \! \left(n \right)}{9+n}+\frac{\left(43 n +60\right) a \! \left(1+n \right)}{9+n}-\frac{3 \left(59 n +146\right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(163 n +573\right) a \! \left(n +3\right)}{9+n}-\frac{3 \left(110 n +503\right) a \! \left(n +4\right)}{9+n}+\frac{\left(197 n +1113\right) a \! \left(n +5\right)}{9+n}-\frac{3 \left(23 n +155\right) a \! \left(n +6\right)}{9+n}+\frac{\left(13 n +102\right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 17 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 17 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)