Av(1243, 1324, 2314, 3412)
Generating Function
\(\displaystyle -\frac{7 x^{7}-34 x^{6}+77 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 189, 516, 1319, 3200, 7456, 16846, 37190, 80696, 172871, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+7 x^{7}-34 x^{6}+77 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 189\)
\(\displaystyle a \! \left(7\right) = 516\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{5}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{24}+\frac{23 n^{2}}{24}+4 a \! \left(n +1\right)-4 a \! \left(n \right)-\frac{19 n}{20}+2, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 189\)
\(\displaystyle a \! \left(7\right) = 516\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{5}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{24}+\frac{23 n^{2}}{24}+4 a \! \left(n +1\right)-4 a \! \left(n \right)-\frac{19 n}{20}+2, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle -\frac{217 n}{60}-3-\frac{3 n^{3}}{8}-\frac{23 n^{2}}{24}-\frac{n^{4}}{24}-\frac{n^{5}}{120}+2^{-1+n} n +4 \,2^{n}\)
This specification was found using the strategy pack "Point And Row Placements" and has 71 rules.
Found on July 23, 2021.Finding the specification took 12 seconds.
Copy 71 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_{17}\! \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_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{21}\! \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_{13}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{17}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{31}\! \left(x \right) &= 0\\
F_{32}\! \left(x \right) &= F_{17}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{17}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{17}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{17}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{17}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{17}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{17}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{17}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{17}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{17}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{17}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{17}\! \left(x \right) F_{66}\! \left(x \right)\\
\end{align*}\)