Av(1234, 1243, 2341, 3412, 4123)
Generating Function
\(\displaystyle -\frac{2 x^{7}+x^{6}+3 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 150, 368, 843, 1838, 3874, 7986, 16235, 32732, 65686, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}+x^{6}+3 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 150\)
\(\displaystyle a \! \left(7\right) = 368\)
\(\displaystyle a \! \left(n +1\right) = -\frac{n^{4}}{12}+\frac{5 n^{3}}{3}-\frac{89 n^{2}}{12}+2 a \! \left(n \right)+\frac{95 n}{6}-12, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 150\)
\(\displaystyle a \! \left(7\right) = 368\)
\(\displaystyle a \! \left(n +1\right) = -\frac{n^{4}}{12}+\frac{5 n^{3}}{3}-\frac{89 n^{2}}{12}+2 a \! \left(n \right)+\frac{95 n}{6}-12, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 3+4 \,2^{n}-\frac{35 n}{3}+\frac{47 n^{2}}{12}-\frac{4 n^{3}}{3}+\frac{n^{4}}{12} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Insertion Point Placements" and has 68 rules.
Found on July 23, 2021.Finding the specification took 8 seconds.
Copy 68 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{29}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{13}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{13}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{11}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{11}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{13}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{13}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{39}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{11} \left(x \right)^{3} F_{10}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{13}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\
\end{align*}\)