Av(1432, 2413, 3214)
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Generating Function
\(\displaystyle \frac{\left(x^{2}-3 x +1\right) \left(-1+x \right)^{5}}{4 x^{7}-23 x^{6}+55 x^{5}-78 x^{4}+66 x^{3}-33 x^{2}+9 x -1}\)
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Counting Sequence
1, 1, 2, 6, 21, 74, 253, 843, 2772, 9080, 29759, 97686, 321033, 1055596, 3471365, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{7}-23 x^{6}+55 x^{5}-78 x^{4}+66 x^{3}-33 x^{2}+9 x -1\right) F \! \left(x \right)-\left(x^{2}-3 x +1\right) \left(-1+x \right)^{5} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(6\right) = 253\)
\(\displaystyle a \! \left(7\right) = 843\)
\(\displaystyle a \! \left(n +7\right) = 4 a \! \left(n \right)-23 a \! \left(n +1\right)+55 a \! \left(n +2\right)-78 a \! \left(n +3\right)+66 a \! \left(n +4\right)-33 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \frac{33479224 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +5}}{43764829}-\frac{172552762 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +4}}{43764829}+\frac{357051072 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +3}}{43764829}-\frac{438342235 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +2}}{43764829}+\frac{286639494 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +1}}{43764829}+\frac{15176275 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n -1}}{43764829}-\frac{96386032 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf} \left(4 Z^{7}-23 Z^{6}+55 Z^{5}-78 Z^{4}+66 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n}}{43764829}+\left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 67 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{20}\! \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_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{50}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{63}\! \left(x \right)\\ \end{align*}\)