Av(1324, 1432, 2413, 3142)
Generating Function
\(\displaystyle -\frac{\left(-3 x^{2}+\left(x^{2}-x +1\right) \sqrt{1-4 x}+3 x -1\right) \left(x -1\right)^{2}}{2 x^{2} \left(x^{3}-x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 225, 764, 2631, 9181, 32402, 115453, 414754, 1500578, 5463000, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{3}-x +1\right) F \left(x
\right)^{2}-\left(3 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 225\)
\(\displaystyle a \! \left(n +7\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}-\frac{\left(47+6 n \right) a \! \left(2+n \right)}{9+n}+\frac{\left(9 n +26\right) a \! \left(n +1\right)}{9+n}-\frac{\left(-15+7 n \right) a \! \left(n +3\right)}{9+n}+\frac{6 \left(3 n +11\right) a \! \left(n +4\right)}{9+n}-\frac{4 \left(4 n +23\right) a \! \left(n +5\right)}{9+n}+\frac{\left(7 n +52\right) a \! \left(n +6\right)}{9+n}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 225\)
\(\displaystyle a \! \left(n +7\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}-\frac{\left(47+6 n \right) a \! \left(2+n \right)}{9+n}+\frac{\left(9 n +26\right) a \! \left(n +1\right)}{9+n}-\frac{\left(-15+7 n \right) a \! \left(n +3\right)}{9+n}+\frac{6 \left(3 n +11\right) a \! \left(n +4\right)}{9+n}-\frac{4 \left(4 n +23\right) a \! \left(n +5\right)}{9+n}+\frac{\left(7 n +52\right) a \! \left(n +6\right)}{9+n}, \quad n \geq 7\)
This specification was found using the strategy pack "Point Placements" and has 17 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 17 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{0}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)