Av(1324, 2143, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(-2 x^{4}+9 x^{3}-10 x^{2}+5 x -1\right) \sqrt{1-4 x}+2 x^{5}+6 x^{4}-11 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 199, 620, 1972, 6443, 21598, 73981, 257824, 910851, 3253088, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}-\left(2 x -1\right) \left(2 x^{5}+6 x^{4}-11 x^{3}+10 x^{2}-5 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+10 x^{8}-39 x^{7}+107 x^{6}-175 x^{5}+175 x^{4}-111 x^{3}+44 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 199\)
\(\displaystyle a \! \left(7\right) = 620\)
\(\displaystyle a \! \left(8\right) = 1972\)
\(\displaystyle a \! \left(9\right) = 6443\)
\(\displaystyle a \! \left(n +7\right) = \frac{8 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{4 \left(42+25 n \right) a \! \left(n +1\right)}{n +8}+\frac{110 \left(5+2 n \right) a \! \left(n +2\right)}{n +8}-\frac{5 \left(170+49 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(716+157 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(336+59 n \right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(41+6 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 n -10}{n +8}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 199\)
\(\displaystyle a \! \left(7\right) = 620\)
\(\displaystyle a \! \left(8\right) = 1972\)
\(\displaystyle a \! \left(9\right) = 6443\)
\(\displaystyle a \! \left(n +7\right) = \frac{8 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{4 \left(42+25 n \right) a \! \left(n +1\right)}{n +8}+\frac{110 \left(5+2 n \right) a \! \left(n +2\right)}{n +8}-\frac{5 \left(170+49 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(716+157 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(336+59 n \right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(41+6 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 n -10}{n +8}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 30 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 30 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_{24}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{20} \left(x \right)^{2}\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{20}\! \left(x \right) F_{5}\! \left(x \right) F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)