Av(1342, 2314, 2341, 4123)
Generating Function
\(\displaystyle \frac{-3 \left(x^{3}-\frac{5}{3} x^{2}+\frac{4}{3} x -\frac{1}{3}\right) \left(-1+x \right)^{5} \sqrt{1-4 x}-4 x^{9}+9 x^{8}-28 x^{7}+67 x^{6}-107 x^{5}+112 x^{4}-78 x^{3}+35 x^{2}-9 x +1}{2 x \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(-1+x \right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 203, 624, 1933, 6119, 19893, 66379, 226618, 788544, 2786577, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(3 x^{3}-5 x^{2}+4 x -1\right)^{2} \left(-1+x \right)^{10} F \left(x
\right)^{2}+\left(4 x^{9}-9 x^{8}+28 x^{7}-67 x^{6}+107 x^{5}-112 x^{4}+78 x^{3}-35 x^{2}+9 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(-1+x \right)^{5} F \! \left(x \right)+4 x^{17}-9 x^{16}-46 x^{15}+524 x^{14}-2443 x^{13}+7279 x^{12}-15551 x^{11}+25134 x^{10}-31624 x^{9}+31434 x^{8}-24808 x^{7}+15497 x^{6}-7576 x^{5}+2837 x^{4}-785 x^{3}+151 x^{2}-18 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 203\)
\(\displaystyle a \! \left(7\right) = 624\)
\(\displaystyle a \! \left(8\right) = 1933\)
\(\displaystyle a \! \left(9\right) = 6119\)
\(\displaystyle a \! \left(10\right) = 19893\)
\(\displaystyle a \! \left(11\right) = 66379\)
\(\displaystyle a \! \left(12\right) = 226618\)
\(\displaystyle a \! \left(13\right) = 788544\)
\(\displaystyle a \! \left(n +7\right) = \frac{18 \left(2 n +1\right) a \! \left(n \right)}{n +8}-\frac{3 \left(43 n +66\right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(113 n +290\right) a \! \left(n +2\right)}{n +8}-\frac{\left(233 n +840\right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(75 n +349\right) a \! \left(n +4\right)}{n +8}-\frac{2 \left(29 n +166\right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(6 n +41\right) a \! \left(n +6\right)}{n +8}-\frac{3 n^{5}-34 n^{4}+205 n^{3}-902 n^{2}+704 n -336}{24 \left(n +8\right)}, \quad n \geq 14\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 203\)
\(\displaystyle a \! \left(7\right) = 624\)
\(\displaystyle a \! \left(8\right) = 1933\)
\(\displaystyle a \! \left(9\right) = 6119\)
\(\displaystyle a \! \left(10\right) = 19893\)
\(\displaystyle a \! \left(11\right) = 66379\)
\(\displaystyle a \! \left(12\right) = 226618\)
\(\displaystyle a \! \left(13\right) = 788544\)
\(\displaystyle a \! \left(n +7\right) = \frac{18 \left(2 n +1\right) a \! \left(n \right)}{n +8}-\frac{3 \left(43 n +66\right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(113 n +290\right) a \! \left(n +2\right)}{n +8}-\frac{\left(233 n +840\right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(75 n +349\right) a \! \left(n +4\right)}{n +8}-\frac{2 \left(29 n +166\right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(6 n +41\right) a \! \left(n +6\right)}{n +8}-\frac{3 n^{5}-34 n^{4}+205 n^{3}-902 n^{2}+704 n -336}{24 \left(n +8\right)}, \quad n \geq 14\)
This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 125 rules.
Found on July 23, 2021.Finding the specification took 14 seconds.
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\(\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_{100}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{28}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 0\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{43}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{49}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{51}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{75}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{84}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{9}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{9}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x , 1\right)\\
F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{114}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{124}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)+F_{85}\! \left(x \right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{110}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{111}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{114}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= y x\\
F_{114}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{111}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= 2 F_{38}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{123}\! \left(x , y\right)\\
F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{117}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{118}\! \left(x , y\right)\\
F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)+F_{38}\! \left(x \right)\\
F_{119}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{120}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{122}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{118}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{107}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{124}\! \left(x , y\right) &= \frac{F_{103}\! \left(x , y\right) y -F_{103}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)