Av(1234, 1342, 2413, 3412)
Generating Function
\(\displaystyle -\frac{6 x^{7}-36 x^{6}+78 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 20, 63, 181, 480, 1199, 2871, 6676, 15210, 34150, 75839, 166967, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+6 x^{7}-36 x^{6}+78 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(7\right) = 480\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{5}}{120}-\frac{n^{4}}{8}+\frac{3 n^{3}}{8}+\frac{5 n^{2}}{8}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{53 n}{60}+2, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(7\right) = 480\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{5}}{120}-\frac{n^{4}}{8}+\frac{3 n^{3}}{8}+\frac{5 n^{2}}{8}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{53 n}{60}+2, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle 5+2^{n} n -4 \,2^{n}+\frac{29 n}{20}+\frac{n^{5}}{120}-\frac{n^{4}}{24}+\frac{n^{3}}{24}+\frac{13 n^{2}}{24}\)
This specification was found using the strategy pack "Point Placements" and has 63 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 63 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
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_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{60}\! \left(x \right)\\
\end{align*}\)