Av(1324, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(x^{5}-2 x^{4}-2 x^{3}+8 x^{2}-5 x +1\right) \sqrt{1-4 x}+2 x^{6}-13 x^{5}+34 x^{4}-42 x^{3}+28 x^{2}-9 x +1}{4 \left(x^{3}-4 x^{2}+5 x -1\right) \left(x -1\right)^{2} \left(x -\frac{1}{2}\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 267, 951, 3407, 12309, 44867, 164891, 610347, 2273020, 8508804, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-4 x^{2}+5 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}-\left(x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+7 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x \left(x^{8}-8 x^{7}+35 x^{6}-81 x^{5}+107 x^{4}-81 x^{3}+36 x^{2}-9 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 267\)
\(\displaystyle a \! \left(7\right) = 951\)
\(\displaystyle a \! \left(8\right) = 3407\)
\(\displaystyle a \! \left(9\right) = 12309\)
\(\displaystyle a \! \left(10\right) = 44867\)
\(\displaystyle a \! \left(11\right) = 164891\)
\(\displaystyle a \! \left(12\right) = 610347\)
\(\displaystyle a \! \left(n +11\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +11}+\frac{2 \left(29+31 n \right) a \! \left(n +1\right)}{n +11}-\frac{\left(178+179 n \right) a \! \left(n +2\right)}{n +11}+\frac{\left(-325+157 n \right) a \! \left(n +3\right)}{n +11}+\frac{\left(3086+347 n \right) a \! \left(n +4\right)}{n +11}-\frac{\left(7849+1158 n \right) a \! \left(n +5\right)}{n +11}+\frac{\left(10461+1498 n \right) a \! \left(n +6\right)}{n +11}-\frac{\left(8231+1084 n \right) a \! \left(n +7\right)}{n +11}+\frac{5 \left(787+94 n \right) a \! \left(n +8\right)}{n +11}-\frac{\left(1117+121 n \right) a \! \left(n +9\right)}{n +11}+\frac{\left(172+17 n \right) a \! \left(n +10\right)}{n +11}-\frac{2}{n +11}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 267\)
\(\displaystyle a \! \left(7\right) = 951\)
\(\displaystyle a \! \left(8\right) = 3407\)
\(\displaystyle a \! \left(9\right) = 12309\)
\(\displaystyle a \! \left(10\right) = 44867\)
\(\displaystyle a \! \left(11\right) = 164891\)
\(\displaystyle a \! \left(12\right) = 610347\)
\(\displaystyle a \! \left(n +11\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +11}+\frac{2 \left(29+31 n \right) a \! \left(n +1\right)}{n +11}-\frac{\left(178+179 n \right) a \! \left(n +2\right)}{n +11}+\frac{\left(-325+157 n \right) a \! \left(n +3\right)}{n +11}+\frac{\left(3086+347 n \right) a \! \left(n +4\right)}{n +11}-\frac{\left(7849+1158 n \right) a \! \left(n +5\right)}{n +11}+\frac{\left(10461+1498 n \right) a \! \left(n +6\right)}{n +11}-\frac{\left(8231+1084 n \right) a \! \left(n +7\right)}{n +11}+\frac{5 \left(787+94 n \right) a \! \left(n +8\right)}{n +11}-\frac{\left(1117+121 n \right) a \! \left(n +9\right)}{n +11}+\frac{\left(172+17 n \right) a \! \left(n +10\right)}{n +11}-\frac{2}{n +11}, \quad n \geq 13\)
This specification was found using the strategy pack "Row And Col Placements" and has 37 rules.
Found on July 23, 2021.Finding the specification took 11 seconds.
Copy 37 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{11}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14} \left(x \right)^{3}\\
F_{32}\! \left(x \right) &= F_{20} \left(x \right)^{2} F_{33}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right) F_{34}\! \left(x \right)\\
\end{align*}\)