Av(1234, 1243, 1342, 3124, 3241)
Generating Function
\(\displaystyle \frac{x^{7}-3 x^{6}+5 x^{5}-7 x^{4}+9 x^{3}-10 x^{2}+5 x -1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 54, 141, 347, 820, 1887, 4270, 9562, 21277, 47166, 104325, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{7}+3 x^{6}-5 x^{5}+7 x^{4}-9 x^{3}+10 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 141\)
\(\displaystyle a \! \left(7\right) = 347\)
\(\displaystyle a \! \left(n \right) = -\frac{n^{3}}{6}-\frac{3 n^{2}}{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-\frac{7 n}{3}-2, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 141\)
\(\displaystyle a \! \left(7\right) = 347\)
\(\displaystyle a \! \left(n \right) = -\frac{n^{3}}{6}-\frac{3 n^{2}}{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-\frac{7 n}{3}-2, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle -\frac{23}{16}+\frac{371 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+2 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{944}+\frac{57 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+2 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{944}+\frac{593 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+2 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{944}-\frac{n^{3}}{12}-\frac{5 n^{2}}{8}-\frac{2 n}{3}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 74 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 74 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_{17}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{47}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{70}\! \left(x \right)\\
\end{align*}\)