Av(1243, 1324, 2413, 4231)
Generating Function
\(\displaystyle \frac{3 x^{9}-20 x^{8}+59 x^{7}-123 x^{6}+177 x^{5}-172 x^{4}+110 x^{3}-44 x^{2}+10 x -1}{\left(2 x -1\right)^{2} \left(-1+x \right)^{7}}\)
Counting Sequence
1, 1, 2, 6, 20, 61, 168, 426, 1014, 2302, 5049, 10810, 22776, 47511, 98546, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(-1+x \right)^{7} F \! \left(x \right)-3 x^{9}+20 x^{8}-59 x^{7}+123 x^{6}-177 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 168\)
\(\displaystyle a \! \left(7\right) = 426\)
\(\displaystyle a \! \left(8\right) = 1014\)
\(\displaystyle a \! \left(9\right) = 2302\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{7 n^{5}}{240}+\frac{23 n^{4}}{144}-\frac{17 n^{3}}{48}-\frac{29 n^{2}}{180}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{203 n}{60}-1, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 168\)
\(\displaystyle a \! \left(7\right) = 426\)
\(\displaystyle a \! \left(8\right) = 1014\)
\(\displaystyle a \! \left(9\right) = 2302\)
\(\displaystyle a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{7 n^{5}}{240}+\frac{23 n^{4}}{144}-\frac{17 n^{3}}{48}-\frac{29 n^{2}}{180}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{203 n}{60}-1, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -2-\frac{4 n}{5}-\frac{3 n^{3}}{16}+\frac{5 n^{4}}{144}-\frac{n^{5}}{80}+\frac{n^{6}}{720}+\frac{9 \,2^{n}}{4}+\frac{2^{n} n}{4}-\frac{373 n^{2}}{360} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 135 rules.
Found on July 23, 2021.Finding the specification took 6 seconds.
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\(\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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{20}\! \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_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{12}\! \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_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{17}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{10}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{10}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 0\\
F_{47}\! \left(x \right) &= F_{10}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{10}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{10}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{59}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{10}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{10}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{10}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{10}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{71}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{10}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{10}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{77}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{10}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{10}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{10}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{10}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{10}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{10}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{46}\! \left(x \right)+F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{10}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{10}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{10}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{10}\! \left(x \right) F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{10}\! \left(x \right) F_{101}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{46}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{10}\! \left(x \right) F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{116}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{10}\! \left(x \right) F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{10}\! \left(x \right) F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{124}\! \left(x \right)+F_{126}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{10}\! \left(x \right) F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{10}\! \left(x \right) F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{10}\! \left(x \right) F_{111}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{10}\! \left(x \right) F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{10}\! \left(x \right) F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{50}\! \left(x \right)\\
\end{align*}\)