Av(1234, 1342, 1423, 3412)
Generating Function
\(\displaystyle -\frac{8 x^{8}-42 x^{7}+111 x^{6}-173 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{7}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 200, 583, 1621, 4329, 11170, 27984, 68350, 163325, 382960, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{7} F \! \left(x \right)+8 x^{8}-42 x^{7}+111 x^{6}-173 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 200\)
\(\displaystyle a \! \left(7\right) = 583\)
\(\displaystyle a \! \left(8\right) = 1621\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{n^{5}}{80}+\frac{17 n^{4}}{144}-\frac{17 n^{3}}{48}+\frac{497 n^{2}}{360}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{17 n}{15}+2, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 200\)
\(\displaystyle a \! \left(7\right) = 583\)
\(\displaystyle a \! \left(8\right) = 1621\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{n^{5}}{80}+\frac{17 n^{4}}{144}-\frac{17 n^{3}}{48}+\frac{497 n^{2}}{360}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{17 n}{15}+2, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle 35+\frac{1151 n}{60}-34 \,2^{n}+\frac{23 n^{4}}{144}+\frac{13 n^{3}}{16}+\frac{n^{5}}{240}+\frac{n^{6}}{720}+4 \,2^{n} n +\frac{1051 n^{2}}{180}\)
This specification was found using the strategy pack "Point Placements" and has 37 rules.
Found on July 23, 2021.Finding the specification took 5 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_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{15}\! \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_{31}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{15}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{30}\! \left(x \right)\\
\end{align*}\)