Av(1243, 1324, 2431, 4132)
Generating Function
\(\displaystyle \frac{-4 \left(x -\frac{1}{2}\right)^{2} \left(x -1\right)^{4} \sqrt{1-4 x}-2 x^{7}+12 x^{6}-36 x^{5}+51 x^{4}-46 x^{3}+26 x^{2}-8 x +1}{4 \left(x -\frac{1}{2}\right) x \left(x -1\right)^{3} \left(x^{2}-3 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 214, 692, 2258, 7476, 25148, 85879, 297278, 1041433, 3686815, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(2 x^{7}-12 x^{6}+36 x^{5}-51 x^{4}+46 x^{3}-26 x^{2}+8 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{13}+4 x^{12}-92 x^{11}+501 x^{10}-1498 x^{9}+2927 x^{8}-3982 x^{7}+3884 x^{6}-2741 x^{5}+1388 x^{4}-491 x^{3}+115 x^{2}-16 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(7\right) = 692\)
\(\displaystyle a \! \left(8\right) = 2258\)
\(\displaystyle a \! \left(9\right) = 7476\)
\(\displaystyle a \! \left(n +6\right) = -\frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{4 \left(21 n +26\right) a \! \left(n +1\right)}{n +7}-\frac{4 \left(38 n +89\right) a \! \left(n +2\right)}{n +7}+\frac{3 \left(43 n +151\right) a \! \left(n +3\right)}{n +7}-\frac{2 \left(28 n +131\right) a \! \left(n +4\right)}{n +7}+\frac{2 \left(6 n +35\right) a \! \left(n +5\right)}{n +7}-\frac{2 \left(3 n^{2}-9 n -10\right)}{n +7}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(7\right) = 692\)
\(\displaystyle a \! \left(8\right) = 2258\)
\(\displaystyle a \! \left(9\right) = 7476\)
\(\displaystyle a \! \left(n +6\right) = -\frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{4 \left(21 n +26\right) a \! \left(n +1\right)}{n +7}-\frac{4 \left(38 n +89\right) a \! \left(n +2\right)}{n +7}+\frac{3 \left(43 n +151\right) a \! \left(n +3\right)}{n +7}-\frac{2 \left(28 n +131\right) a \! \left(n +4\right)}{n +7}+\frac{2 \left(6 n +35\right) a \! \left(n +5\right)}{n +7}-\frac{2 \left(3 n^{2}-9 n -10\right)}{n +7}, \quad n \geq 10\)
This specification was found using the strategy pack "Row And Col Placements" and has 44 rules.
Found on January 17, 2022.Finding the specification took 16 seconds.
Copy 44 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_{28}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{4} \left(x \right)^{2}\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \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) F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{5}\! \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_{12}\! \left(x \right) F_{32}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{40}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{30}\! \left(x \right) F_{7}\! \left(x \right)\\
\end{align*}\)