Av(1234, 1243, 2143, 2431, 4213)
Generating Function
\(\displaystyle \frac{2 x^{14}+8 x^{13}+7 x^{12}-10 x^{11}-21 x^{10}-3 x^{9}+22 x^{8}+7 x^{7}-18 x^{6}-6 x^{5}+3 x^{4}+5 x^{3}+x^{2}-3 x +1}{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 110, 240, 513, 1066, 2169, 4342, 8581, 16784, 32552, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+2 x^{14}+8 x^{13}+7 x^{12}-10 x^{11}-21 x^{10}-3 x^{9}+22 x^{8}+7 x^{7}-18 x^{6}-6 x^{5}+3 x^{4}+5 x^{3}+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 110\)
\(\displaystyle a \! \left(7\right) = 240\)
\(\displaystyle a \! \left(8\right) = 513\)
\(\displaystyle a \! \left(9\right) = 1066\)
\(\displaystyle a \! \left(10\right) = 2169\)
\(\displaystyle a \! \left(11\right) = 4342\)
\(\displaystyle a \! \left(12\right) = 8581\)
\(\displaystyle a \! \left(13\right) = 16784\)
\(\displaystyle a \! \left(14\right) = 32552\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(2+n \right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}-\frac{5}{4}, \quad n \geq 15\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 110\)
\(\displaystyle a \! \left(7\right) = 240\)
\(\displaystyle a \! \left(8\right) = 513\)
\(\displaystyle a \! \left(9\right) = 1066\)
\(\displaystyle a \! \left(10\right) = 2169\)
\(\displaystyle a \! \left(11\right) = 4342\)
\(\displaystyle a \! \left(12\right) = 8581\)
\(\displaystyle a \! \left(13\right) = 16784\)
\(\displaystyle a \! \left(14\right) = 32552\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(2+n \right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}-\frac{5}{4}, \quad n \geq 15\)
Explicit Closed Form
\(\displaystyle -\frac{\left(n +1\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-2-n}\right)}{5}-\frac{64 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{25}-\frac{132 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{25}-\frac{28 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{11}-\frac{19 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{22}+\frac{\left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{11}+\frac{64 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{11}+\left(\left\{\begin{array}{cc}1 & n =1\text{ or } n =4 \\ 4 & n =5 \\ 2 & n =6 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 72 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 72 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_{24}\! \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_{18}\! \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_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \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_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \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_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{53}\! \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_{48}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
\end{align*}\)