Av(1243, 1432, 2413)
Counting Sequence
1, 1, 2, 6, 21, 76, 279, 1043, 3979, 15464, 61035, 243956, 985332, 4015149, 16486978, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x -1\right)^{2} F \left(x
\right)^{3}-x \left(3 x^{2}-4 x +2\right) F \left(x
\right)^{2}+\left(x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 279\)
\(\displaystyle a \! \left(7\right) = 1043\)
\(\displaystyle a \! \left(8\right) = 3979\)
\(\displaystyle a \! \left(9\right) = 15464\)
\(\displaystyle a \! \left(10\right) = 61035\)
\(\displaystyle a \! \left(11\right) = 243956\)
\(\displaystyle a \! \left(12\right) = 985332\)
\(\displaystyle a \! \left(n +13\right) = -\frac{6 \left(n +2\right) \left(2 n +1\right) a \! \left(n \right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(44 n^{2}+110 n +69\right) a \! \left(n +1\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(160 n^{2}+1926 n +3869\right) a \! \left(n +2\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(5273 n^{2}+50735 n +114018\right) a \! \left(n +3\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(11181 n^{2}+120515 n +317741\right) a \! \left(n +4\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(51779 n^{2}+639265 n +1956368\right) a \! \left(n +5\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(38713 n^{2}+545045 n +1910803\right) a \! \left(n +6\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{2 \left(40080 n^{2}+637090 n +2525749\right) a \! \left(n +7\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(29561 n^{2}+524608 n +2323625\right) a \! \left(n +8\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{4 \left(7783 n^{2}+152573 n +746767\right) a \! \left(n +9\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(5727 n^{2}+122867 n +658286\right) a \! \left(n +10\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(2791 n^{2}+65025 n +378356\right) a \! \left(n +11\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{4 \left(20 n +243\right) a \! \left(n +12\right)}{5 \left(n +14\right)}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 279\)
\(\displaystyle a \! \left(7\right) = 1043\)
\(\displaystyle a \! \left(8\right) = 3979\)
\(\displaystyle a \! \left(9\right) = 15464\)
\(\displaystyle a \! \left(10\right) = 61035\)
\(\displaystyle a \! \left(11\right) = 243956\)
\(\displaystyle a \! \left(12\right) = 985332\)
\(\displaystyle a \! \left(n +13\right) = -\frac{6 \left(n +2\right) \left(2 n +1\right) a \! \left(n \right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(44 n^{2}+110 n +69\right) a \! \left(n +1\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(160 n^{2}+1926 n +3869\right) a \! \left(n +2\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(5273 n^{2}+50735 n +114018\right) a \! \left(n +3\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(11181 n^{2}+120515 n +317741\right) a \! \left(n +4\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(51779 n^{2}+639265 n +1956368\right) a \! \left(n +5\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(38713 n^{2}+545045 n +1910803\right) a \! \left(n +6\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{2 \left(40080 n^{2}+637090 n +2525749\right) a \! \left(n +7\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(29561 n^{2}+524608 n +2323625\right) a \! \left(n +8\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{4 \left(7783 n^{2}+152573 n +746767\right) a \! \left(n +9\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{2 \left(5727 n^{2}+122867 n +658286\right) a \! \left(n +10\right)}{25 \left(n +14\right) \left(n +13\right)}-\frac{\left(2791 n^{2}+65025 n +378356\right) a \! \left(n +11\right)}{25 \left(n +14\right) \left(n +13\right)}+\frac{4 \left(20 n +243\right) a \! \left(n +12\right)}{5 \left(n +14\right)}, \quad n \geq 13\)
This specification was found using the strategy pack "Point Placements" and has 21 rules.
Found on July 23, 2021.Finding the specification took 6 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{0}\! \left(x \right) F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\
\end{align*}\)