Av(1243, 1324, 2431)
Generating Function
\(\displaystyle \frac{\left(-8 x^{5}+26 x^{4}-37 x^{3}+25 x^{2}-8 x +1\right) \sqrt{1-4 x}-4 x^{6}+28 x^{5}-66 x^{4}+73 x^{3}-39 x^{2}+10 x -1}{4 \left(x -1\right)^{2} x^{3} \left(x -\frac{1}{2}\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 283, 1032, 3740, 13522, 48930, 177564, 646908, 2367121, 8699706, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(4 x^{6}-28 x^{5}+66 x^{4}-73 x^{3}+39 x^{2}-10 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+4 x^{9}+8 x^{8}-104 x^{7}+302 x^{6}-452 x^{5}+403 x^{4}-220 x^{3}+72 x^{2}-13 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 283\)
\(\displaystyle a \! \left(7\right) = 1032\)
\(\displaystyle a \! \left(8\right) = 3740\)
\(\displaystyle a \! \left(9\right) = 13522\)
\(\displaystyle a \! \left(n +8\right) = -\frac{32 \left(2 n +1\right) a \! \left(n \right)}{11+n}+\frac{2 \left(454 n +1935\right) a \! \left(3+n \right)}{11+n}+\frac{8 \left(40 n +67\right) a \! \left(n +1\right)}{11+n}-\frac{4 \left(179 n +532\right) a \! \left(n +2\right)}{11+n}-\frac{3 \left(233 n +1294\right) a \! \left(n +4\right)}{11+n}+\frac{\left(2281+332 n \right) a \! \left(n +5\right)}{11+n}-\frac{\left(95 n +781\right) a \! \left(n +6\right)}{11+n}+\frac{3 \left(5 n +48\right) a \! \left(n +7\right)}{11+n}+\frac{4}{11+n}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 283\)
\(\displaystyle a \! \left(7\right) = 1032\)
\(\displaystyle a \! \left(8\right) = 3740\)
\(\displaystyle a \! \left(9\right) = 13522\)
\(\displaystyle a \! \left(n +8\right) = -\frac{32 \left(2 n +1\right) a \! \left(n \right)}{11+n}+\frac{2 \left(454 n +1935\right) a \! \left(3+n \right)}{11+n}+\frac{8 \left(40 n +67\right) a \! \left(n +1\right)}{11+n}-\frac{4 \left(179 n +532\right) a \! \left(n +2\right)}{11+n}-\frac{3 \left(233 n +1294\right) a \! \left(n +4\right)}{11+n}+\frac{\left(2281+332 n \right) a \! \left(n +5\right)}{11+n}-\frac{\left(95 n +781\right) a \! \left(n +6\right)}{11+n}+\frac{3 \left(5 n +48\right) a \! \left(n +7\right)}{11+n}+\frac{4}{11+n}, \quad n \geq 10\)
This specification was found using the strategy pack "Row And Col Placements" and has 47 rules.
Found on July 23, 2021.Finding the specification took 57 seconds.
Copy 47 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_{7}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4} \left(x \right)^{2} F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4} \left(x \right)^{2}\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{17}\! \left(x \right) F_{21}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{21}\! \left(x \right) F_{28}\! \left(x \right) F_{32}\! \left(x \right) F_{35}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{28}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{32}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{39}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4} \left(x \right)^{5} F_{7}\! \left(x \right)\\
\end{align*}\)