Av(2413, 14253, 31425, 42531)
Generating Function
\(\displaystyle \frac{\left(-x^{4}+7 x^{3}-13 x^{2}+6 x -1\right) \sqrt{x^{2}-6 x +1}-x^{5}+10 x^{4}-30 x^{3}+29 x^{2}-7 x +1}{2 x}\)
Counting Sequence
1, 1, 2, 6, 23, 100, 464, 2236, 11048, 55588, 283648, 1463868, 7626296, 40049188, 211768752, ...
Implicit Equation for the Generating Function
\(\displaystyle x F \left(x
\right)^{2}+\left(x^{5}-10 x^{4}+30 x^{3}-29 x^{2}+7 x -1\right) F \! \left(x \right)+3 x^{4}-14 x^{3}+22 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 23\)
\(\displaystyle a \! \left(5\right) = 100\)
\(\displaystyle a \! \left(n +6\right) = -\frac{\left(n -4\right) a \! \left(n \right)}{7+n}+\frac{\left(-29+13 n \right) a \! \left(1+n \right)}{7+n}-\frac{2 \left(-11+28 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(136+91 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(169+50 n \right) a \! \left(n +4\right)}{7+n}+\frac{3 \left(21+4 n \right) a \! \left(n +5\right)}{7+n}, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 23\)
\(\displaystyle a \! \left(5\right) = 100\)
\(\displaystyle a \! \left(n +6\right) = -\frac{\left(n -4\right) a \! \left(n \right)}{7+n}+\frac{\left(-29+13 n \right) a \! \left(1+n \right)}{7+n}-\frac{2 \left(-11+28 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(136+91 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(169+50 n \right) a \! \left(n +4\right)}{7+n}+\frac{3 \left(21+4 n \right) a \! \left(n +5\right)}{7+n}, \quad n \geq 6\)
This specification was found using the strategy pack "Row And Col Placements Req Corrob Expand Verified" and has 51 rules.
Found on January 22, 2022.Finding the specification took 20 seconds.
Copy 51 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_{8}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{3}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right) F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{37}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{21}\! \left(x \right) F_{38}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{31}\! \left(x \right) F_{45}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{48}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{48}\! \left(x \right) &= \frac{F_{49}\! \left(x \right)}{F_{11}\! \left(x \right) F_{8}\! \left(x \right)}\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= -F_{38}\! \left(x \right)+F_{37}\! \left(x \right)\\
\end{align*}\)