Av(1234, 1342, 1432, 3124, 3142)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(-1+x \right)^{3}}{x^{4}+4 x^{3}-5 x^{2}+4 x -1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 55, 151, 411, 1128, 3116, 8619, 23819, 65773, 181589, 501386, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(-1+x \right)^{3} = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+4 a \! \left(n +1\right)-5 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{\left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{170}+\frac{\left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{170}+\frac{13 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{85}+\frac{11 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{170}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 53 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 53 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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \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_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{22}\! \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_{15}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{30}\! \left(x \right)\\ \end{align*}\)