Av(1342, 2143, 2413, 4132)
Generating Function
\(\displaystyle \frac{-3 \left(x^{2}-x +\frac{1}{3}\right) \left(x -1\right)^{3} \sqrt{1-4 x}+4 x^{6}-5 x^{5}+11 x^{3}-13 x^{2}+6 x -1}{16 x^{6}-44 x^{5}+58 x^{4}-40 x^{3}+14 x^{2}-2 x}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 216, 711, 2370, 8013, 27467, 95347, 334745, 1187040, 4246525, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) \left(2 x -1\right)^{2} F \left(x
\right)^{2}-\left(2 x -1\right) \left(4 x^{6}-5 x^{5}+11 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-15 x^{5}+37 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 216\)
\(\displaystyle a \! \left(7\right) = 711\)
\(\displaystyle a \! \left(8\right) = 2370\)
\(\displaystyle a \! \left(9\right) = 8013\)
\(\displaystyle a \! \left(n +9\right) = \frac{48 \left(1+2 n \right) a \! \left(n \right)}{n +10}-\frac{12 \left(71+40 n \right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(1581+559 n \right) a \! \left(n +2\right)}{n +10}-\frac{\left(6014+1571 n \right) a \! \left(n +3\right)}{n +10}+\frac{6 \left(1164+241 n \right) a \! \left(n +4\right)}{n +10}-\frac{\left(5232+895 n \right) a \! \left(n +5\right)}{n +10}+\frac{2 \left(1271+185 n \right) a \! \left(n +6\right)}{n +10}-\frac{2 \left(387+49 n \right) a \! \left(n +7\right)}{n +10}+\frac{\left(134+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 216\)
\(\displaystyle a \! \left(7\right) = 711\)
\(\displaystyle a \! \left(8\right) = 2370\)
\(\displaystyle a \! \left(9\right) = 8013\)
\(\displaystyle a \! \left(n +9\right) = \frac{48 \left(1+2 n \right) a \! \left(n \right)}{n +10}-\frac{12 \left(71+40 n \right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(1581+559 n \right) a \! \left(n +2\right)}{n +10}-\frac{\left(6014+1571 n \right) a \! \left(n +3\right)}{n +10}+\frac{6 \left(1164+241 n \right) a \! \left(n +4\right)}{n +10}-\frac{\left(5232+895 n \right) a \! \left(n +5\right)}{n +10}+\frac{2 \left(1271+185 n \right) a \! \left(n +6\right)}{n +10}-\frac{2 \left(387+49 n \right) a \! \left(n +7\right)}{n +10}+\frac{\left(134+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 26 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 26 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
\end{align*}\)