Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 23154, 23451, 23541, 24153, 24351, 34152, 34251)
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Counting Sequence
1, 1, 2, 6, 24, 105, 477, 2229, 10663, 51994, 257513, 1291929, 6551930, 33534282, 172998311, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} x^{2} F \left(x \right)^{3}+x \left(3 x^{3}-16 x^{2}+23 x -14\right) F \left(x \right)^{2}-\left(x -2\right) \left(4 x^{3}-5 x^{2}+6 x +2\right) F \! \left(x \right)+x^{4}-2 x^{3}+2 x^{2}-4 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 105\)
\(\displaystyle a \! \left(6\right) = 477\)
\(\displaystyle a \! \left(7\right) = 2229\)
\(\displaystyle a \! \left(8\right) = 10663\)
\(\displaystyle a \! \left(9\right) = 51994\)
\(\displaystyle a \! \left(n +10\right) = -\frac{343 n \left(n +1\right) a \! \left(n \right)}{26 \left(n +11\right) \left(n +10\right)}+\frac{49 \left(n +1\right) \left(11 n +7\right) a \! \left(n +1\right)}{13 \left(n +11\right) \left(n +10\right)}-\frac{7 \left(218 n^{2}-679 n -3390\right) a \! \left(n +2\right)}{65 \left(n +11\right) \left(n +10\right)}-\frac{\left(3359 n^{2}+33691 n +85457\right) a \! \left(n +3\right)}{65 \left(n +11\right) \left(n +10\right)}+\frac{\left(3210 n^{2}+16133 n +10061\right) a \! \left(n +4\right)}{65 \left(n +11\right) \left(n +10\right)}+\frac{\left(5451 n^{2}+92684 n +352916\right) a \! \left(n +5\right)}{65 \left(n +11\right) \left(n +10\right)}-\frac{\left(12329 n^{2}+188598 n +712975\right) a \! \left(n +6\right)}{65 \left(n +11\right) \left(n +10\right)}+\frac{\left(10151 n^{2}+164543 n +665898\right) a \! \left(n +7\right)}{65 \left(n +11\right) \left(n +10\right)}-\frac{\left(8465 n^{2}+149275 n +657716\right) a \! \left(n +8\right)}{130 \left(n +11\right) \left(n +10\right)}+\frac{2 \left(431 n +4007\right) a \! \left(n +9\right)}{65 \left(n +11\right)}, \quad n \geq 10\)
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This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 20 rules.

Found on January 23, 2022.

Finding the specification took 15 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= \frac{F_{5}\! \left(x , y\right) y -F_{5}\! \left(x , 1\right)}{-1+y}\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)^{2} F_{5}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 21 rules.

Found on January 22, 2022.

Finding the specification took 22 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , 1, y\right)\\ F_{11}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y , z\right)+F_{17}\! \left(x , y , z\right)+F_{19}\! \left(x , z , y\right)\\ F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , y , z\right)\\ F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , z\right) F_{3}\! \left(x \right) F_{7}\! \left(x , z\right)\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right)\\ F_{18}\! \left(x , y , z\right) &= F_{10}\! \left(x , z\right) F_{14}\! \left(x , z\right) F_{9}\! \left(x , z\right)\\ F_{19}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , z\right)\\ F_{20}\! \left(x , y , z\right) &= F_{14}\! \left(x , y\right)^{2} F_{5}\! \left(x , z\right) F_{9}\! \left(x , y\right)\\ \end{align*}\)