Av(1342, 2413, 3412)
Generating Function
\(\displaystyle \frac{4 \left(x -1\right)^{3} \left(x -\frac{1}{2}\right)^{2} \sqrt{1-4 x}-12 x^{6}+64 x^{5}-114 x^{4}+101 x^{3}-47 x^{2}+11 x -1}{8 x^{7}-64 x^{6}+192 x^{5}-276 x^{4}+218 x^{3}-96 x^{2}+22 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 286, 1068, 4006, 15093, 57104, 216875, 826448, 3158726, 12104591, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{7}-32 x^{6}+96 x^{5}-138 x^{4}+109 x^{3}-48 x^{2}+11 x -1\right) F \left(x
\right)^{2}+\left(12 x^{6}-64 x^{5}+114 x^{4}-101 x^{3}+47 x^{2}-11 x +1\right) F \! \left(x \right)+x \left(9 x^{4}-20 x^{3}+18 x^{2}-7 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(7\right) = 1068\)
\(\displaystyle a \! \left(8\right) = 4006\)
\(\displaystyle a \! \left(9\right) = 15093\)
\(\displaystyle a \! \left(n +10\right) = -\frac{16 \left(2 n +3\right) a \! \left(n \right)}{n +10}+\frac{8 \left(39 n +82\right) a \! \left(n +1\right)}{n +10}-\frac{4 \left(311 n +886\right) a \! \left(n +2\right)}{n +10}+\frac{4 \left(669 n +2488\right) a \! \left(n +3\right)}{n +10}-\frac{4 \left(877 n +4049\right) a \! \left(n +4\right)}{n +10}+\frac{4 \left(743 n +4100\right) a \! \left(n +5\right)}{n +10}-\frac{\left(10662+1661 n \right) a \! \left(n +6\right)}{n +10}+\frac{\left(607 n +4442\right) a \! \left(n +7\right)}{n +10}-\frac{\left(1142+139 n \right) a \! \left(n +8\right)}{n +10}+\frac{2 \left(9 n +82\right) a \! \left(n +9\right)}{n +10}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(7\right) = 1068\)
\(\displaystyle a \! \left(8\right) = 4006\)
\(\displaystyle a \! \left(9\right) = 15093\)
\(\displaystyle a \! \left(n +10\right) = -\frac{16 \left(2 n +3\right) a \! \left(n \right)}{n +10}+\frac{8 \left(39 n +82\right) a \! \left(n +1\right)}{n +10}-\frac{4 \left(311 n +886\right) a \! \left(n +2\right)}{n +10}+\frac{4 \left(669 n +2488\right) a \! \left(n +3\right)}{n +10}-\frac{4 \left(877 n +4049\right) a \! \left(n +4\right)}{n +10}+\frac{4 \left(743 n +4100\right) a \! \left(n +5\right)}{n +10}-\frac{\left(10662+1661 n \right) a \! \left(n +6\right)}{n +10}+\frac{\left(607 n +4442\right) a \! \left(n +7\right)}{n +10}-\frac{\left(1142+139 n \right) a \! \left(n +8\right)}{n +10}+\frac{2 \left(9 n +82\right) a \! \left(n +9\right)}{n +10}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 27 rules.
Found on July 23, 2021.Finding the specification took 11 seconds.
Copy 27 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 0\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\
\end{align*}\)