Av(12435, 12453, 12543, 14235, 14253, 14325, 21435, 21453, 21543, 24135, 24153, 24315, 24351, 24513, 24531, 25143, 25413, 25431, 41235, 41253, 41325, 42135, 42153, 42315, 42351, 42513, 42531, 43125, 43215, 43251)
Generating Function
\(\displaystyle \frac{4 x^{7}+6 x^{6}-25 x^{5}+37 x^{4}-32 x^{3}+20 x^{2}-7 x +1}{\left(4 x^{3}-5 x^{2}+4 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 300, 938, 2864, 8658, 25940, 76978, 226440, 661258, 1919484, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(4 x^{3}-5 x^{2}+4 x -1\right)^{2} F \! \left(x \right)+4 x^{7}+6 x^{6}-25 x^{5}+37 x^{4}-32 x^{3}+20 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 300\)
\(\displaystyle a(7) = 938\)
\(\displaystyle a{\left(n + 6 \right)} = - 16 a{\left(n \right)} + 40 a{\left(n + 1 \right)} - 57 a{\left(n + 2 \right)} + 48 a{\left(n + 3 \right)} - 26 a{\left(n + 4 \right)} + 8 a{\left(n + 5 \right)}, \quad n \geq 8\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 300\)
\(\displaystyle a(7) = 938\)
\(\displaystyle a{\left(n + 6 \right)} = - 16 a{\left(n \right)} + 40 a{\left(n + 1 \right)} - 57 a{\left(n + 2 \right)} + 48 a{\left(n + 3 \right)} - 26 a{\left(n + 4 \right)} + 8 a{\left(n + 5 \right)}, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ -\frac{13 n \left(\left(\left(\left(-\frac{17 i}{299}+\frac{200 \sqrt{29}}{8671}\right) \sqrt{3}+\frac{600 i \sqrt{29}}{8671}-\frac{17}{299}\right) \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{322}{13}+\left(\left(i-\frac{4 \sqrt{29}}{377}\right) \sqrt{3}+\frac{12 i \sqrt{29}}{377}-1\right) \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(19 i-12 \sqrt{29}\right) \sqrt{3}-36 i \sqrt{29}+19\right) \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{12696}-\frac{i \sqrt{3}\, \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{5}{12}\right)^{-n}+\left(\left(\left(\frac{17 i}{299}+\frac{200 \sqrt{29}}{8671}\right) \sqrt{3}-\frac{600 i \sqrt{29}}{8671}-\frac{17}{299}\right) \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{322}{13}+\left(\left(-i-\frac{4 \sqrt{29}}{377}\right) \sqrt{3}-\frac{12 i \sqrt{29}}{377}-1\right) \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-19 i-12 \sqrt{29}\right) \sqrt{3}+36 i \sqrt{29}+19\right) \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{12696}+\frac{i \sqrt{3}\, \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{5}{12}\right)^{-n}+\frac{8 \left(-\frac{\left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{5}{12}-\frac{19 \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{6348}+\frac{\left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{29}\, \sqrt{3}}{529}\right)^{-n} \left(\left(-\frac{50 \sqrt{29}\, \sqrt{3}}{23}+\frac{493}{92}\right) \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}+\sqrt{3}\, \sqrt{29}\, \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{377 \left(19+12 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{4}-\frac{4669}{4}\right)}{377}\right)}{2208} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 142 rules.
Finding the specification took 141 seconds.
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Copy 142 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_{14}\! \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_{33}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{14}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{14}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{14}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{14}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{14}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{14}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{14}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{14}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{84}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{85}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{14}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{88}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{85}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= 0\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{14}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{94}\! \left(x \right) &= 0\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{14}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{104}\! \left(x \right)+F_{18}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{104}\! \left(x \right) &= 0\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{127}\! \left(x \right)+F_{18}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{127}\! \left(x \right) &= 0\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{18}\! \left(x \right)+F_{72}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{129}\! \left(x \right)+F_{18}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{134}\! \left(x \right)+F_{141}\! \left(x \right)+F_{18}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{137}\! \left(x \right)\\
F_{141}\! \left(x \right) &= 0\\
\end{align*}\)