Av(1243, 1324, 1342, 2431)
Generating Function
\(\displaystyle \frac{\left(-x^{6}+3 x^{5}-3 x^{4}-5 x^{3}+9 x^{2}-5 x +1\right) \sqrt{1-4 x}+3 x^{6}-15 x^{5}+17 x^{4}-3 x^{3}-7 x^{2}+5 x -1}{2 x \left(2 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 221, 725, 2394, 8011, 27222, 93880, 328053, 1159448, 4137936, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}-\left(2 x -1\right) \left(3 x^{6}-15 x^{5}+17 x^{4}-3 x^{3}-7 x^{2}+5 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{12}-4 x^{11}-6 x^{10}+70 x^{9}-169 x^{8}+188 x^{7}-50 x^{6}-135 x^{5}+199 x^{4}-135 x^{3}+52 x^{2}-11 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 221\)
\(\displaystyle a \! \left(7\right) = 725\)
\(\displaystyle a \! \left(8\right) = 2394\)
\(\displaystyle a \! \left(9\right) = 8011\)
\(\displaystyle a \! \left(10\right) = 27222\)
\(\displaystyle a \! \left(11\right) = 93880\)
\(\displaystyle a \! \left(12\right) = 328053\)
\(\displaystyle a \! \left(n +9\right) = -\frac{4 \left(2 n -1\right) a \! \left(n \right)}{n +10}+\frac{2 \left(19 n +4\right) a \! \left(n +1\right)}{n +10}-\frac{\left(73 n +67\right) a \! \left(n +2\right)}{n +10}+\frac{3 \left(8 n -49\right) a \! \left(n +3\right)}{n +10}+\frac{\left(787+118 n \right) a \! \left(n +4\right)}{n +10}-\frac{\left(198 n +1255\right) a \! \left(n +5\right)}{n +10}+\frac{\left(1023+146 n \right) a \! \left(n +6\right)}{n +10}-\frac{\left(58 n +459\right) a \! \left(n +7\right)}{n +10}+\frac{\left(107+12 n \right) a \! \left(n +8\right)}{n +10}+\frac{n^{2}-n -4}{n +10}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 221\)
\(\displaystyle a \! \left(7\right) = 725\)
\(\displaystyle a \! \left(8\right) = 2394\)
\(\displaystyle a \! \left(9\right) = 8011\)
\(\displaystyle a \! \left(10\right) = 27222\)
\(\displaystyle a \! \left(11\right) = 93880\)
\(\displaystyle a \! \left(12\right) = 328053\)
\(\displaystyle a \! \left(n +9\right) = -\frac{4 \left(2 n -1\right) a \! \left(n \right)}{n +10}+\frac{2 \left(19 n +4\right) a \! \left(n +1\right)}{n +10}-\frac{\left(73 n +67\right) a \! \left(n +2\right)}{n +10}+\frac{3 \left(8 n -49\right) a \! \left(n +3\right)}{n +10}+\frac{\left(787+118 n \right) a \! \left(n +4\right)}{n +10}-\frac{\left(198 n +1255\right) a \! \left(n +5\right)}{n +10}+\frac{\left(1023+146 n \right) a \! \left(n +6\right)}{n +10}-\frac{\left(58 n +459\right) a \! \left(n +7\right)}{n +10}+\frac{\left(107+12 n \right) a \! \left(n +8\right)}{n +10}+\frac{n^{2}-n -4}{n +10}, \quad n \geq 13\)
This specification was found using the strategy pack "Point Placements" and has 40 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 40 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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{0}\! \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_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 0\\
F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{25}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{19}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{25}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)