Av(1243, 1342, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(-5 x^{7}+23 x^{6}-50 x^{5}+66 x^{4}-55 x^{3}+28 x^{2}-8 x +1\right) \sqrt{1-4 x}-4 x^{8}+x^{7}+x^{6}+12 x^{5}-36 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{16 x^{8}-76 x^{7}+162 x^{6}-200 x^{5}+152 x^{4}-70 x^{3}+18 x^{2}-2 x}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 222, 734, 2445, 8246, 28192, 97648, 342220, 1211852, 4330431, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(4 x^{8}-x^{7}-x^{6}-12 x^{5}+36 x^{4}-43 x^{3}+26 x^{2}-8 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{11}+8 x^{10}-44 x^{9}+148 x^{8}-332 x^{7}+508 x^{6}-541 x^{5}+399 x^{4}-199 x^{3}+64 x^{2}-12 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 222\)
\(\displaystyle a \! \left(7\right) = 734\)
\(\displaystyle a \! \left(8\right) = 2445\)
\(\displaystyle a \! \left(9\right) = 8246\)
\(\displaystyle a \! \left(10\right) = 28192\)
\(\displaystyle a \! \left(11\right) = 97648\)
\(\displaystyle a \! \left(12\right) = 342220\)
\(\displaystyle a \! \left(13\right) = 1211852\)
\(\displaystyle a \! \left(14\right) = 4330431\)
\(\displaystyle a \! \left(n +11\right) = \frac{80 \left(2 n +1\right) a \! \left(n \right)}{12+n}-\frac{4 \left(224 n +335\right) a \! \left(1+n \right)}{12+n}+\frac{2 \left(1273 n +3261\right) a \! \left(n +2\right)}{12+n}-\frac{\left(16960+4643 n \right) a \! \left(n +3\right)}{12+n}+\frac{\left(5911 n +27976\right) a \! \left(n +4\right)}{12+n}-\frac{2 \left(2714 n +15715\right) a \! \left(n +5\right)}{12+n}+\frac{\left(3619 n +24714\right) a \! \left(n +6\right)}{12+n}-\frac{2 \left(867 n +6812\right) a \! \left(n +7\right)}{12+n}+\frac{\left(581 n +5160\right) a \! \left(n +8\right)}{12+n}-\frac{3 \left(43 n +426\right) a \! \left(n +9\right)}{12+n}+\frac{\left(17 n +186\right) a \! \left(n +10\right)}{12+n}+\frac{3 n^{2}-16 n +6}{12+n}, \quad n \geq 15\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 222\)
\(\displaystyle a \! \left(7\right) = 734\)
\(\displaystyle a \! \left(8\right) = 2445\)
\(\displaystyle a \! \left(9\right) = 8246\)
\(\displaystyle a \! \left(10\right) = 28192\)
\(\displaystyle a \! \left(11\right) = 97648\)
\(\displaystyle a \! \left(12\right) = 342220\)
\(\displaystyle a \! \left(13\right) = 1211852\)
\(\displaystyle a \! \left(14\right) = 4330431\)
\(\displaystyle a \! \left(n +11\right) = \frac{80 \left(2 n +1\right) a \! \left(n \right)}{12+n}-\frac{4 \left(224 n +335\right) a \! \left(1+n \right)}{12+n}+\frac{2 \left(1273 n +3261\right) a \! \left(n +2\right)}{12+n}-\frac{\left(16960+4643 n \right) a \! \left(n +3\right)}{12+n}+\frac{\left(5911 n +27976\right) a \! \left(n +4\right)}{12+n}-\frac{2 \left(2714 n +15715\right) a \! \left(n +5\right)}{12+n}+\frac{\left(3619 n +24714\right) a \! \left(n +6\right)}{12+n}-\frac{2 \left(867 n +6812\right) a \! \left(n +7\right)}{12+n}+\frac{\left(581 n +5160\right) a \! \left(n +8\right)}{12+n}-\frac{3 \left(43 n +426\right) a \! \left(n +9\right)}{12+n}+\frac{\left(17 n +186\right) a \! \left(n +10\right)}{12+n}+\frac{3 n^{2}-16 n +6}{12+n}, \quad n \geq 15\)
This specification was found using the strategy pack "Point Placements" and has 28 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 28 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_{12}\! \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_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18} \left(x \right)^{2}\\
\end{align*}\)