Av(1243, 2413, 4132)
Generating Function
\(\displaystyle \frac{-4 \left(x^{2}-3 x +1\right) \left(x^{2}-x +\frac{1}{2}\right)^{2} \left(x -1\right) \sqrt{1-4 x}-6 x^{8}+48 x^{7}-140 x^{6}+210 x^{5}-194 x^{4}+115 x^{3}-44 x^{2}+10 x -1}{2 x^{2} \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right) \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 76, 275, 991, 3566, 12848, 46426, 168390, 613252, 2242584, 8233836, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right) \left(x -1\right)^{2} F \left(x
\right)^{2}+\left(x -1\right) \left(6 x^{8}-48 x^{7}+140 x^{6}-210 x^{5}+194 x^{4}-115 x^{3}+44 x^{2}-10 x +1\right) F \! \left(x \right)+9 x^{8}-56 x^{7}+145 x^{6}-209 x^{5}+191 x^{4}-114 x^{3}+44 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 275\)
\(\displaystyle a \! \left(7\right) = 991\)
\(\displaystyle a \! \left(8\right) = 3566\)
\(\displaystyle a \! \left(9\right) = 12848\)
\(\displaystyle a \! \left(10\right) = 46426\)
\(\displaystyle a \! \left(11\right) = 168390\)
\(\displaystyle a \! \left(12\right) = 613252\)
\(\displaystyle a \! \left(n +11\right) = \frac{4 \left(2 n +3\right) a \! \left(n \right)}{13+n}+\frac{4 \left(125 n +389\right) a \! \left(2+n \right)}{13+n}-\frac{2 \left(49 n +111\right) a \! \left(n +1\right)}{13+n}-\frac{5 \left(279 n +1135\right) a \! \left(n +3\right)}{13+n}+\frac{\left(12273+2399 n \right) a \! \left(n +4\right)}{13+n}-\frac{4 \left(688 n +4279\right) a \! \left(n +5\right)}{13+n}+\frac{2 \left(1091 n +8026\right) a \! \left(n +6\right)}{13+n}-\frac{18 \left(67 n +570\right) a \! \left(n +7\right)}{13+n}+\frac{8 \left(57 n +550\right) a \! \left(n +8\right)}{13+n}-\frac{2 \left(56 n +603\right) a \! \left(n +9\right)}{13+n}+\frac{2 \left(8 n +95\right) a \! \left(n +10\right)}{13+n}+\frac{3 n +9}{13+n}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 275\)
\(\displaystyle a \! \left(7\right) = 991\)
\(\displaystyle a \! \left(8\right) = 3566\)
\(\displaystyle a \! \left(9\right) = 12848\)
\(\displaystyle a \! \left(10\right) = 46426\)
\(\displaystyle a \! \left(11\right) = 168390\)
\(\displaystyle a \! \left(12\right) = 613252\)
\(\displaystyle a \! \left(n +11\right) = \frac{4 \left(2 n +3\right) a \! \left(n \right)}{13+n}+\frac{4 \left(125 n +389\right) a \! \left(2+n \right)}{13+n}-\frac{2 \left(49 n +111\right) a \! \left(n +1\right)}{13+n}-\frac{5 \left(279 n +1135\right) a \! \left(n +3\right)}{13+n}+\frac{\left(12273+2399 n \right) a \! \left(n +4\right)}{13+n}-\frac{4 \left(688 n +4279\right) a \! \left(n +5\right)}{13+n}+\frac{2 \left(1091 n +8026\right) a \! \left(n +6\right)}{13+n}-\frac{18 \left(67 n +570\right) a \! \left(n +7\right)}{13+n}+\frac{8 \left(57 n +550\right) a \! \left(n +8\right)}{13+n}-\frac{2 \left(56 n +603\right) a \! \left(n +9\right)}{13+n}+\frac{2 \left(8 n +95\right) a \! \left(n +10\right)}{13+n}+\frac{3 n +9}{13+n}, \quad n \geq 13\)
This specification was found using the strategy pack "Point Placements" and has 32 rules.
Found on January 17, 2022.Finding the specification took 8 seconds.
Copy 32 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_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{9}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{24}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{23}\! \left(x \right) F_{5}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)