Av(1324, 1342, 2314, 4231)
Generating Function
\(\displaystyle \frac{x^{7}-6 x^{6}+19 x^{5}-39 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 203, 612, 1791, 5114, 14315, 39445, 107370, 289532, 775182, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{7}+6 x^{6}-19 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 203\)
\(\displaystyle a \! \left(7\right) = 612\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{3}}{6}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{n}{6}+2, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 203\)
\(\displaystyle a \! \left(7\right) = 612\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{3}}{6}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{n}{6}+2, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(9 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-9 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}+\frac{11 n}{6}-\\7 \,2^{n -1}+3 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Row Placements" and has 64 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 64 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_{9}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= 2 F_{4}\! \left(x \right)+F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \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) &= x\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{21}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{24}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= 0\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{27}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{40}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{43}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{10}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{48}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{29}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{50}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right) F_{43}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{36}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{63}\! \left(x \right)+F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{62}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)