Av(1234, 1324, 1342, 1423, 2341, 4123)
Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+5 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{4}+2 x^{3}-5 x^{2}+4 x -1}{4 x \left(x -\frac{1}{2}\right) \left(-1+x \right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 18, 53, 158, 486, 1550, 5109, 17298, 59799, 210048, 746983, 2682618, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(-1+x \right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{4}-2 x^{3}+5 x^{2}-4 x +1\right) \left(-1+x \right)^{2} F \! \left(x \right)+x^{7}+2 x^{6}-15 x^{5}+37 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{4+n}-\frac{4 \left(5 n +8\right) a \! \left(1+n \right)}{4+n}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{4+n}+\frac{n \left(3 n -13\right)}{4+n}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{4+n}-\frac{4 \left(5 n +8\right) a \! \left(1+n \right)}{4+n}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{4+n}+\frac{n \left(3 n -13\right)}{4+n}, \quad n \geq 7\)
This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 73 rules.
Found on July 23, 2021.Finding the specification took 12 seconds.
Copy 73 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{1}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{3}\! \left(x \right)\\
F_{2}\! \left(x \right) &= 1\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= x\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{23}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{3}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{3}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{34}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\
F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{53}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{53}\! \left(x , y\right) F_{56}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= y x\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{44}\! \left(x \right)+F_{59}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{62}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= 2 F_{16}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{16}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{56}\! \left(x , y\right) F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{3}\! \left(x \right)+F_{66}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{59}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= \frac{F_{51}\! \left(x , y\right) y -F_{51}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)