Av(1234, 1342, 2143, 3124)
Generating Function
\(\displaystyle -\frac{2 x^{4}+x^{3}-2 x^{2}+3 x -1}{2 x^{5}-3 x^{4}-3 x^{3}+4 x^{2}-4 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 63, 194, 598, 1853, 5751, 17842, 55329, 171564, 532013, 1649807, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-3 x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)+2 x^{4}+x^{3}-2 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)+3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)+3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{19054 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{1165027}+\frac{19054 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{1165027}+\frac{19054 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{1165027}+\frac{19054 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{1165027}+\frac{19054 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{1165027}+\frac{52041 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{1165027}+\frac{52041 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{1165027}+\frac{52041 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{1165027}+\frac{52041 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{1165027}+\frac{52041 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{1165027}+\frac{42916 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{1165027}+\frac{42916 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{1165027}+\frac{42916 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{1165027}+\frac{42916 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{1165027}+\frac{42916 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{1165027}+\frac{36100 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{1165027}+\frac{36100 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{1165027}+\frac{36100 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{1165027}+\frac{36100 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{1165027}+\frac{36100 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{1165027}+\frac{120888 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}}{1165027}+\frac{120888 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}}{1165027}+\frac{120888 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}}{1165027}+\frac{120888 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}}{1165027}+\frac{120888 \mathit{RootOf} \left(2 Z^{5}-3 Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{1165027}\)
This specification was found using the strategy pack "Point Placements" and has 59 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 59 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \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_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \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_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{55}\! \left(x \right)\\
\end{align*}\)