Av(13452, 13524, 13542, 14352, 31452, 31524, 31542)
Counting Sequence
1, 1, 2, 6, 24, 113, 581, 3149, 17688, 102001, 600303, 3590921, 21768532, 133438243, 825696844, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) F \left(x
\right)^{4}+\left(-x +2\right) F \left(x
\right)^{3}+\left(-x -1\right) F \left(x
\right)^{2}+F \! \left(x \right)-1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 113\)
\(\displaystyle a \! \left(6\right) = 581\)
\(\displaystyle a \! \left(7\right) = 3149\)
\(\displaystyle a \! \left(8\right) = 17688\)
\(\displaystyle a \! \left(9\right) = 102001\)
\(\displaystyle a \! \left(10\right) = 600303\)
\(\displaystyle a \! \left(11\right) = 3590921\)
\(\displaystyle a \! \left(n +12\right) = \frac{1600 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{40 \left(n +1\right) \left(1814 n^{2}+4123 n +2403\right) a \! \left(n +1\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{30 \left(2178 n^{3}-12697 n^{2}-81881 n -96050\right) a \! \left(n +2\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(376384 n^{3}+3722439 n^{2}+12425549 n +13912914\right) a \! \left(n +3\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{12 \left(532131 n^{3}+7201629 n^{2}+32240951 n +47807794\right) a \! \left(n +4\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{4 \left(4749038 n^{3}+75272808 n^{2}+397041913 n +697149048\right) a \! \left(n +5\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{6 \left(4978800 n^{3}+90328153 n^{2}+546634963 n +1103500758\right) a \! \left(n +6\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(28463783 n^{3}+583249173 n^{2}+3987133012 n +9093976596\right) a \! \left(n +7\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{\left(17194963 n^{3}+393821559 n^{2}+3007073090 n +7655825904\right) a \! \left(n +8\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(3278329 n^{3}+83200071 n^{2}+703342100 n +1980771132\right) a \! \left(n +9\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{2 \left(750095 n^{2}+13435303 n +60158832\right) a \! \left(n +10\right)}{8897 \left(n +12\right) \left(n +11\right)}+\frac{\left(182641 n +1732056\right) a \! \left(n +11\right)}{8897 n +106764}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 113\)
\(\displaystyle a \! \left(6\right) = 581\)
\(\displaystyle a \! \left(7\right) = 3149\)
\(\displaystyle a \! \left(8\right) = 17688\)
\(\displaystyle a \! \left(9\right) = 102001\)
\(\displaystyle a \! \left(10\right) = 600303\)
\(\displaystyle a \! \left(11\right) = 3590921\)
\(\displaystyle a \! \left(n +12\right) = \frac{1600 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{40 \left(n +1\right) \left(1814 n^{2}+4123 n +2403\right) a \! \left(n +1\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{30 \left(2178 n^{3}-12697 n^{2}-81881 n -96050\right) a \! \left(n +2\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(376384 n^{3}+3722439 n^{2}+12425549 n +13912914\right) a \! \left(n +3\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{12 \left(532131 n^{3}+7201629 n^{2}+32240951 n +47807794\right) a \! \left(n +4\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{4 \left(4749038 n^{3}+75272808 n^{2}+397041913 n +697149048\right) a \! \left(n +5\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{6 \left(4978800 n^{3}+90328153 n^{2}+546634963 n +1103500758\right) a \! \left(n +6\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(28463783 n^{3}+583249173 n^{2}+3987133012 n +9093976596\right) a \! \left(n +7\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{\left(17194963 n^{3}+393821559 n^{2}+3007073090 n +7655825904\right) a \! \left(n +8\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(3278329 n^{3}+83200071 n^{2}+703342100 n +1980771132\right) a \! \left(n +9\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{2 \left(750095 n^{2}+13435303 n +60158832\right) a \! \left(n +10\right)}{8897 \left(n +12\right) \left(n +11\right)}+\frac{\left(182641 n +1732056\right) a \! \left(n +11\right)}{8897 n +106764}, \quad n \geq 12\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 20 rules.
Found on January 25, 2022.Finding the specification took 1272 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= \frac{F_{10}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{11}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
\end{align*}\)