Av(13452, 14253, 14352, 15243, 15342, 23451, 24351, 25341)
Generating Function
\(\displaystyle \frac{-\left(2 x -1\right)^{\frac{5}{2}} x \left(x^{4}-10 x^{3}+13 x^{2}-6 x +1\right) \sqrt{-1+6 x}+70 x^{8}-640 x^{7}+1868 x^{6}-2603 x^{5}+2020 x^{4}-923 x^{3}+246 x^{2}-35 x +2}{214 x^{8}-1268 x^{7}+2924 x^{6}-3520 x^{5}+2470 x^{4}-1048 x^{3}+264 x^{2}-36 x +2}\)
Counting Sequence
1, 1, 2, 6, 24, 112, 562, 2920, 15468, 83013, 450036, 2460313, 13546668, 75047600, 417966986, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(107 x^{8}-634 x^{7}+1462 x^{6}-1760 x^{5}+1235 x^{4}-524 x^{3}+132 x^{2}-18 x +1\right) F \left(x
\right)^{2}+\left(-70 x^{8}+640 x^{7}-1868 x^{6}+2603 x^{5}-2020 x^{4}+923 x^{3}-246 x^{2}+35 x -2\right) F \! \left(x \right)+11 x^{8}-134 x^{7}+542 x^{6}-915 x^{5}+804 x^{4}-401 x^{3}+114 x^{2}-17 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 562\)
\(\displaystyle a(7) = 2920\)
\(\displaystyle a(8) = 15468\)
\(\displaystyle a(9) = 83013\)
\(\displaystyle a(10) = 450036\)
\(\displaystyle a(11) = 2460313\)
\(\displaystyle a(12) = 13546668\)
\(\displaystyle a(13) = 75047600\)
\(\displaystyle a{\left(n + 14 \right)} = - \frac{1284 n a{\left(n \right)}}{n + 13} + \frac{4 \left(8 n + 97\right) a{\left(n + 13 \right)}}{n + 13} - \frac{\left(457 n + 5133\right) a{\left(n + 12 \right)}}{n + 13} + \frac{4 \left(968 n + 9995\right) a{\left(n + 11 \right)}}{n + 13} + \frac{8 \left(2663 n + 3531\right) a{\left(n + 1 \right)}}{n + 13} - \frac{43 \left(2885 n + 6453\right) a{\left(n + 2 \right)}}{n + 13} - \frac{4 \left(5448 n + 51265\right) a{\left(n + 10 \right)}}{n + 13} + \frac{4 \left(21562 n + 183049\right) a{\left(n + 9 \right)}}{n + 13} - \frac{12 \left(67035 n + 384664\right) a{\left(n + 6 \right)}}{n + 13} + \frac{4 \left(94604 n + 292491\right) a{\left(n + 3 \right)}}{n + 13} + \frac{4 \left(130148 n + 865421\right) a{\left(n + 7 \right)}}{n + 13} + \frac{4 \left(225300 n + 1090187\right) a{\left(n + 5 \right)}}{n + 13} - \frac{3 \left(237443 n + 939291\right) a{\left(n + 4 \right)}}{n + 13} - \frac{\left(247321 n + 1871721\right) a{\left(n + 8 \right)}}{n + 13}, \quad n \geq 14\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 562\)
\(\displaystyle a(7) = 2920\)
\(\displaystyle a(8) = 15468\)
\(\displaystyle a(9) = 83013\)
\(\displaystyle a(10) = 450036\)
\(\displaystyle a(11) = 2460313\)
\(\displaystyle a(12) = 13546668\)
\(\displaystyle a(13) = 75047600\)
\(\displaystyle a{\left(n + 14 \right)} = - \frac{1284 n a{\left(n \right)}}{n + 13} + \frac{4 \left(8 n + 97\right) a{\left(n + 13 \right)}}{n + 13} - \frac{\left(457 n + 5133\right) a{\left(n + 12 \right)}}{n + 13} + \frac{4 \left(968 n + 9995\right) a{\left(n + 11 \right)}}{n + 13} + \frac{8 \left(2663 n + 3531\right) a{\left(n + 1 \right)}}{n + 13} - \frac{43 \left(2885 n + 6453\right) a{\left(n + 2 \right)}}{n + 13} - \frac{4 \left(5448 n + 51265\right) a{\left(n + 10 \right)}}{n + 13} + \frac{4 \left(21562 n + 183049\right) a{\left(n + 9 \right)}}{n + 13} - \frac{12 \left(67035 n + 384664\right) a{\left(n + 6 \right)}}{n + 13} + \frac{4 \left(94604 n + 292491\right) a{\left(n + 3 \right)}}{n + 13} + \frac{4 \left(130148 n + 865421\right) a{\left(n + 7 \right)}}{n + 13} + \frac{4 \left(225300 n + 1090187\right) a{\left(n + 5 \right)}}{n + 13} - \frac{3 \left(237443 n + 939291\right) a{\left(n + 4 \right)}}{n + 13} - \frac{\left(247321 n + 1871721\right) a{\left(n + 8 \right)}}{n + 13}, \quad n \geq 14\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 60 rules.
Finding the specification took 4988 seconds.
Copy 60 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_{41}\! \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_{41}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= y x\\
F_{19}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{2}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right) F_{41}\! \left(x \right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= -\frac{-F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{40}\! \left(x , y\right) &= -\frac{-F_{27}\! \left(x , y\right) y +F_{27}\! \left(x , 1\right)}{-1+y}\\
F_{41}\! \left(x \right) &= x\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{41}\! \left(x \right) F_{44}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{2}\! \left(x \right) F_{50}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{41}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{41}\! \left(x \right) F_{50}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{41}\! \left(x \right) F_{57}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= -\frac{-y F_{58}\! \left(x , y\right)+F_{58}\! \left(x , 1\right)}{-1+y}\\
F_{59}\! \left(x , y\right) &= F_{41}\! \left(x \right) F_{58}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
\end{align*}\)