Av(1234, 1243, 1432, 2413, 3214)
Generating Function
\(\displaystyle -\frac{1}{x^{7}+5 x^{6}+10 x^{5}+8 x^{4}+3 x^{3}+x^{2}+x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 49, 117, 297, 784, 2046, 5248, 13456, 34718, 89728, 231475, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}+5 x^{6}+10 x^{5}+8 x^{4}+3 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 49\)
\(\displaystyle a \! \left(6\right) = 117\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)+5 a \! \left(n +1\right)+10 a \! \left(n +2\right)+8 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 49\)
\(\displaystyle a \! \left(6\right) = 117\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)+5 a \! \left(n +1\right)+10 a \! \left(n +2\right)+8 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{12981 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =6\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =7\right)^{-n +5}}{1724975}-\frac{12512 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =6\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =7\right)^{-n +4}}{1724975}+\frac{70451 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =6\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =7\right)^{-n +3}}{1724975}+\frac{422623 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =6\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =7\right)^{-n +2}}{3449950}+\frac{42237 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =6\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =7\right)^{-n +1}}{344995}+\frac{174179 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =6\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =7\right)^{-n -1}}{3449950}+\frac{351739 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =6\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf} \left(Z^{7}+5 Z^{6}+10 Z^{5}+8 Z^{4}+3 Z^{3}+Z^{2}+Z -1, \mathit{index} =7\right)^{-n}}{1724975}\)
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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{19}\! \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_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
\end{align*}\)