Av(1432, 2143, 2413, 3142)
Generating Function
\(\displaystyle \frac{2 x^{3}-3 x^{2}+3 x -1+\sqrt{-4 x^{7}+28 x^{6}-72 x^{5}+101 x^{4}-82 x^{3}+39 x^{2}-10 x +1}}{2 x \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 226, 777, 2733, 9814, 35847, 132760, 497330, 1881091, 7174068, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{3} F \left(x
\right)^{2}-\left(2 x -1\right) \left(x^{2}-x +1\right) F \! \left(x \right)+\left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 226\)
\(\displaystyle a \! \left(7\right) = 777\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{2 \left(25+16 n \right) a \! \left(1+n \right)}{9+n}-\frac{4 \left(63+25 n \right) a \! \left(n +2\right)}{9+n}+\frac{\left(613+173 n \right) a \! \left(n +3\right)}{9+n}-\frac{\left(836+183 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(681+121 n \right) a \! \left(n +5\right)}{9+n}-\frac{7 \left(47+7 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(86+11 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 226\)
\(\displaystyle a \! \left(7\right) = 777\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{2 \left(25+16 n \right) a \! \left(1+n \right)}{9+n}-\frac{4 \left(63+25 n \right) a \! \left(n +2\right)}{9+n}+\frac{\left(613+173 n \right) a \! \left(n +3\right)}{9+n}-\frac{\left(836+183 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(681+121 n \right) a \! \left(n +5\right)}{9+n}-\frac{7 \left(47+7 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(86+11 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 21 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\
\end{align*}\)