Av(1243, 1342, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(-3 x^{4}+9 x^{3}-10 x^{2}+5 x -1\right) \sqrt{1-4 x}+7 x^{4}-11 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(-1+x \right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 206, 651, 2085, 6818, 22791, 77729, 269656, 948742, 3376701, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(-1+x \right)^{6} F \left(x
\right)^{2}-\left(2 x -1\right) \left(7 x^{4}-11 x^{3}+10 x^{2}-5 x +1\right) \left(-1+x \right)^{3} F \! \left(x \right)+9 x^{8}-44 x^{7}+116 x^{6}-180 x^{5}+176 x^{4}-111 x^{3}+44 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 206\)
\(\displaystyle a \! \left(7\right) = 651\)
\(\displaystyle a \! \left(n +5\right) = \frac{12 \left(2 n +1\right) a \! \left(n \right)}{6+n}-\frac{6 \left(11 n +15\right) a \! \left(1+n \right)}{6+n}+\frac{\left(71 n +174\right) a \! \left(n +2\right)}{6+n}-\frac{2 \left(19 n +69\right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(5 n +24\right) a \! \left(n +4\right)}{6+n}-\frac{\left(3 n -4\right) \left(n -3\right)}{6+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 206\)
\(\displaystyle a \! \left(7\right) = 651\)
\(\displaystyle a \! \left(n +5\right) = \frac{12 \left(2 n +1\right) a \! \left(n \right)}{6+n}-\frac{6 \left(11 n +15\right) a \! \left(1+n \right)}{6+n}+\frac{\left(71 n +174\right) a \! \left(n +2\right)}{6+n}-\frac{2 \left(19 n +69\right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(5 n +24\right) a \! \left(n +4\right)}{6+n}-\frac{\left(3 n -4\right) \left(n -3\right)}{6+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Row Placements" and has 57 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 57 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_{13}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{13}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{15}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{26}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{13}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{13}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 0\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{13}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{13}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{13}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{13}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{13}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{13}\! \left(x \right) F_{35}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
\end{align*}\)