Av(1243, 1342, 1423, 2413, 4231)
View Raw Data
Generating Function
\(\displaystyle -\frac{6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 57, 160, 424, 1073, 2619, 6214, 14418, 32863, 73837, 163964, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{n}{2}+2, \quad n \geq 5\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle 5+\frac{n^{2}}{2}+\frac{3 n}{2}+2^{n} n -4 \,2^{n}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 83 rules.

Found on July 23, 2021.

Finding the specification took 7 seconds.

This tree is too big to show here. Click to view tree on new page.

Copy 83 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_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{20}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{21}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \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)^{2} F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= x\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right) F_{20}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{37}\! \left(x \right) &= 0\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{20}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{20}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{20}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{20}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{20}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{20}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{20}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{20}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{20}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{20}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{20}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{20}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{20}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\ \end{align*}\)