Av(1342, 2413, 3241)
Generating Function
\(\displaystyle \frac{-4 \left(x^{2}-x +1\right) \left(x -1\right)^{2} \left(x -\frac{1}{2}\right)^{2} \sqrt{1-4 x}-2 x^{7}+18 x^{6}-22 x^{5}-7 x^{4}+35 x^{3}-28 x^{2}+9 x -1}{2 x^{7}-16 x^{6}+76 x^{5}-146 x^{4}+144 x^{3}-76 x^{2}+20 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 286, 1069, 4018, 15182, 57636, 219701, 840422, 3224664, 12405795, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-8 x^{6}+38 x^{5}-73 x^{4}+72 x^{3}-38 x^{2}+10 x -1\right) F \left(x
\right)^{2}+\left(2 x^{7}-18 x^{6}+22 x^{5}+7 x^{4}-35 x^{3}+28 x^{2}-9 x +1\right) F \! \left(x \right)+x^{2} \left(x^{5}+6 x^{4}-19 x^{3}+22 x^{2}-11 x +2\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(7\right) = 1069\)
\(\displaystyle a \! \left(8\right) = 4018\)
\(\displaystyle a \! \left(9\right) = 15182\)
\(\displaystyle a \! \left(10\right) = 57636\)
\(\displaystyle a \! \left(11\right) = 219701\)
\(\displaystyle a \! \left(12\right) = 840422\)
\(\displaystyle a \! \left(n +12\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +12}+\frac{2 \left(43 n +49\right) a \! \left(n +1\right)}{n +12}-\frac{\left(831+509 n \right) a \! \left(n +2\right)}{n +12}+\frac{3 \left(558 n +1391\right) a \! \left(n +3\right)}{n +12}-\frac{3 \left(1156 n +4081\right) a \! \left(n +4\right)}{n +12}+\frac{\left(4906 n +22841\right) a \! \left(n +5\right)}{n +12}-\frac{\left(28563+4922 n \right) a \! \left(n +6\right)}{n +12}+\frac{13 \left(272 n +1885\right) a \! \left(n +7\right)}{n +12}-\frac{\left(14410+1797 n \right) a \! \left(n +8\right)}{n +12}+\frac{\left(625 n +5667\right) a \! \left(n +9\right)}{n +12}-\frac{\left(1411+140 n \right) a \! \left(n +10\right)}{n +12}+\frac{\left(18 n +199\right) a \! \left(n +11\right)}{n +12}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(7\right) = 1069\)
\(\displaystyle a \! \left(8\right) = 4018\)
\(\displaystyle a \! \left(9\right) = 15182\)
\(\displaystyle a \! \left(10\right) = 57636\)
\(\displaystyle a \! \left(11\right) = 219701\)
\(\displaystyle a \! \left(12\right) = 840422\)
\(\displaystyle a \! \left(n +12\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +12}+\frac{2 \left(43 n +49\right) a \! \left(n +1\right)}{n +12}-\frac{\left(831+509 n \right) a \! \left(n +2\right)}{n +12}+\frac{3 \left(558 n +1391\right) a \! \left(n +3\right)}{n +12}-\frac{3 \left(1156 n +4081\right) a \! \left(n +4\right)}{n +12}+\frac{\left(4906 n +22841\right) a \! \left(n +5\right)}{n +12}-\frac{\left(28563+4922 n \right) a \! \left(n +6\right)}{n +12}+\frac{13 \left(272 n +1885\right) a \! \left(n +7\right)}{n +12}-\frac{\left(14410+1797 n \right) a \! \left(n +8\right)}{n +12}+\frac{\left(625 n +5667\right) a \! \left(n +9\right)}{n +12}-\frac{\left(1411+140 n \right) a \! \left(n +10\right)}{n +12}+\frac{\left(18 n +199\right) a \! \left(n +11\right)}{n +12}, \quad n \geq 13\)
This specification was found using the strategy pack "Point Placements" and has 25 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right) F_{2}\! \left(x \right) F_{22}\! \left(x \right)\\
\end{align*}\)