Av(12453, 13452, 14352, 15342, 23451, 24351, 25341, 34251, 35241, 45231)
Counting Sequence
1, 1, 2, 6, 24, 110, 530, 2597, 12796, 63156, 311826, 1539461, 7598492, 37496186, 184997956, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(5 x -1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{4}+4 x^{2} \left(5 x -1\right)^{2} \left(x -1\right)^{5} F \left(x
\right)^{3}+\left(5 x -1\right) \left(10 x^{5}-29 x^{4}+50 x^{3}-44 x^{2}+12 x -1\right) \left(x -1\right)^{3} F \left(x
\right)^{2}+2 \left(5 x -1\right) \left(10 x^{5}-49 x^{4}+74 x^{3}-48 x^{2}+12 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+25 x^{8}-295 x^{6}+724 x^{5}-778 x^{4}+434 x^{3}-126 x^{2}+18 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 110\)
\(\displaystyle a{\left(n + 5 \right)} = \frac{350 \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{3 \left(n + 3\right) \left(n + 6\right)} - \frac{4 \left(9 n + 7\right)}{\left(n + 3\right) \left(n + 6\right)} + \frac{2 \left(25 n^{2} + 183 n + 296\right) a{\left(n + 4 \right)}}{3 \left(n + 3\right) \left(n + 6\right)} + \frac{10 \left(99 n^{2} + 382 n + 309\right) a{\left(n + 2 \right)}}{3 \left(n + 3\right) \left(n + 6\right)} - \frac{2 \left(161 n^{2} + 898 n + 1082\right) a{\left(n + 3 \right)}}{3 \left(n + 3\right) \left(n + 6\right)} - \frac{5 \left(283 n^{2} + 661 n + 346\right) a{\left(n + 1 \right)}}{3 \left(n + 3\right) \left(n + 6\right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 110\)
\(\displaystyle a{\left(n + 5 \right)} = \frac{350 \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{3 \left(n + 3\right) \left(n + 6\right)} - \frac{4 \left(9 n + 7\right)}{\left(n + 3\right) \left(n + 6\right)} + \frac{2 \left(25 n^{2} + 183 n + 296\right) a{\left(n + 4 \right)}}{3 \left(n + 3\right) \left(n + 6\right)} + \frac{10 \left(99 n^{2} + 382 n + 309\right) a{\left(n + 2 \right)}}{3 \left(n + 3\right) \left(n + 6\right)} - \frac{2 \left(161 n^{2} + 898 n + 1082\right) a{\left(n + 3 \right)}}{3 \left(n + 3\right) \left(n + 6\right)} - \frac{5 \left(283 n^{2} + 661 n + 346\right) a{\left(n + 1 \right)}}{3 \left(n + 3\right) \left(n + 6\right)}, \quad n \geq 6\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 116 rules.
Finding the specification took 1257 seconds.
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Copy 116 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_{14}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
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_{14}\! \left(x \right) F_{17}\! \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_{14}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{10}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{89}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= -F_{80}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{44}\! \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_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 0\\
F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{48}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{11}\! \left(x \right) F_{14}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{57}\! \left(x \right) &= -F_{63}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{11}\! \left(x \right) F_{14}\! \left(x \right)}\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{48} \left(x \right)^{2} F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{49}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{18}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{14}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{48}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{48}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{17}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{67}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{18}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{17}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{80}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= \frac{F_{82}\! \left(x \right)}{F_{11}\! \left(x \right) F_{14}\! \left(x \right)}\\
F_{82}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{27}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{14}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= \frac{F_{92}\! \left(x \right)}{F_{10}\! \left(x \right)}\\
F_{92}\! \left(x \right) &= -F_{112}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= \frac{F_{94}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{18}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= -F_{109}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= \frac{F_{100}\! \left(x \right)}{F_{14}\! \left(x \right) F_{48}\! \left(x \right)}\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{17}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{106}\! \left(x \right) &= \frac{F_{107}\! \left(x \right)}{F_{11}\! \left(x \right) F_{14}\! \left(x \right)}\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{14}\! \left(x \right) F_{51}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{26}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{115}\! \left(x \right) &= -F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 16 rules.
Finding the specification took 1 seconds.
Copy 16 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= -\frac{-F_{5}\! \left(x , y\right) y +F_{5}\! \left(x , 1\right)}{-1+y}\\
F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , 1, y\right)\\
F_{11}\! \left(x , y , z\right) &= -\frac{-F_{12}\! \left(x , y z \right) y +F_{12}\! \left(x , z\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\
F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)^{2} F_{9}\! \left(x , y\right)\\
\end{align*}\)