Av(2413, 41352, 415263, 531642)
Counting Sequence
1, 1, 2, 6, 23, 102, 492, 2498, 13130, 70800, 389446, 2176802, 12328552, 70597568, 408061604, ...
Implicit Equation for the Generating Function
\(\displaystyle F \left(x
\right)^{5}-4 F \left(x
\right)^{4}+6 F \left(x
\right)^{3}-3 F \left(x
\right)^{2}+\left(x -2\right) F \! \left(x \right)+2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 23\)
\(\displaystyle a \! \left(5\right) = 102\)
\(\displaystyle a \! \left(6\right) = 492\)
\(\displaystyle a \! \left(7\right) = 2498\)
\(\displaystyle a \! \left(8\right) = 13130\)
\(\displaystyle a \! \left(9\right) = 70800\)
\(\displaystyle a \! \left(n +10\right) = \frac{216 n \left(4 n +5\right) \left(2 n +1\right) \left(4 n -1\right) a \! \left(n \right)}{4577975 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{36 \left(6368 n^{4}+7792 n^{3}-18886 n^{2}-31977 n -12042\right) a \! \left(n +1\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{4 \left(470816 n^{4}+8357744 n^{3}+38681974 n^{2}+69859891 n +44181282\right) a \! \left(n +2\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{\left(43164643 n^{4}+575126450 n^{3}+2846628929 n^{2}+6215661292 n +5058417432\right) a \! \left(n +3\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(557299991 n^{4}+8588700937 n^{3}+49566860404 n^{2}+126880219526 n +121477172112\right) a \! \left(n +4\right)}{45779750 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{\left(691999860 n^{4}+12983326459 n^{3}+91058985384 n^{2}+282856686983 n +328212136590\right) a \! \left(n +5\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(1295392235 n^{4}+28657906627 n^{3}+238994266489 n^{2}+891923004767 n +1258904957070\right) a \! \left(n +6\right)}{45779750 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{3 \left(2547757467 n^{3}+54035474921 n^{2}+385270020154 n +921657568480\right) a \! \left(n +7\right)}{183119000 \left(n +8\right) \left(n +9\right) \left(n +10\right)}-\frac{\left(2771667903 n^{2}+43784936273 n +173158009350\right) a \! \left(n +8\right)}{91559500 \left(n +9\right) \left(n +10\right)}+\frac{3 \left(6252531 n +53053816\right) a \! \left(n +9\right)}{1831190 \left(n +10\right)}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 23\)
\(\displaystyle a \! \left(5\right) = 102\)
\(\displaystyle a \! \left(6\right) = 492\)
\(\displaystyle a \! \left(7\right) = 2498\)
\(\displaystyle a \! \left(8\right) = 13130\)
\(\displaystyle a \! \left(9\right) = 70800\)
\(\displaystyle a \! \left(n +10\right) = \frac{216 n \left(4 n +5\right) \left(2 n +1\right) \left(4 n -1\right) a \! \left(n \right)}{4577975 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{36 \left(6368 n^{4}+7792 n^{3}-18886 n^{2}-31977 n -12042\right) a \! \left(n +1\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{4 \left(470816 n^{4}+8357744 n^{3}+38681974 n^{2}+69859891 n +44181282\right) a \! \left(n +2\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{\left(43164643 n^{4}+575126450 n^{3}+2846628929 n^{2}+6215661292 n +5058417432\right) a \! \left(n +3\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(557299991 n^{4}+8588700937 n^{3}+49566860404 n^{2}+126880219526 n +121477172112\right) a \! \left(n +4\right)}{45779750 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{\left(691999860 n^{4}+12983326459 n^{3}+91058985384 n^{2}+282856686983 n +328212136590\right) a \! \left(n +5\right)}{22889875 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(1295392235 n^{4}+28657906627 n^{3}+238994266489 n^{2}+891923004767 n +1258904957070\right) a \! \left(n +6\right)}{45779750 \left(n +10\right) \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{3 \left(2547757467 n^{3}+54035474921 n^{2}+385270020154 n +921657568480\right) a \! \left(n +7\right)}{183119000 \left(n +8\right) \left(n +9\right) \left(n +10\right)}-\frac{\left(2771667903 n^{2}+43784936273 n +173158009350\right) a \! \left(n +8\right)}{91559500 \left(n +9\right) \left(n +10\right)}+\frac{3 \left(6252531 n +53053816\right) a \! \left(n +9\right)}{1831190 \left(n +10\right)}, \quad n \geq 10\)
This specification was found using the strategy pack "Point And Row And Col Placements Expand Verified" and has 33 rules.
Found on January 22, 2022.Finding the specification took 43 seconds.
Copy 33 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_{11}\! \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_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \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_{31}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{29}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Req Corrob Expand Verified" and has 36 rules.
Found on January 22, 2022.Finding the specification took 46 seconds.
Copy 36 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_{7}\! \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_{0}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{0}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{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_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{0}\! \left(x \right) F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{7}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{7}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{20}\! \left(x \right) F_{33}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{28}\! \left(x \right) F_{7}\! \left(x \right)\\
\end{align*}\)