Av(12354, 12453, 13254, 13452, 14253, 14352, 15243, 15342, 21354, 21453, 23154, 23451, 24153, 24351, 25143, 25341, 31254, 31452, 32154, 32451, 34152, 34251, 35142, 35241, 41253, 41352, 42153, 42351, 43152, 43251)
Generating Function
\(\displaystyle \frac{4 x^{5}-2 x^{4}+3 x^{3}-7 x^{2}+5 x -1}{8 x^{3}-11 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 324, 1146, 4032, 14178, 49884, 175602, 618312, 2177322, 7667316, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(-8 x^{3}+11 x^{2}-6 x +1\right) F \! \left(x \right)+4 x^{5}-2 x^{4}+3 x^{3}-7 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a{\left(n + 3 \right)} = 8 a{\left(n \right)} - 11 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a{\left(n + 3 \right)} = 8 a{\left(n \right)} - 11 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(20033 i+552 \sqrt{26}\right) \sqrt{3}-1656 i \sqrt{26}-20033\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}+233818+\left(\left(-4901 i+720 \sqrt{26}\right) \sqrt{3}+2160 i \sqrt{26}-4901\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(181 i-24 \sqrt{26}\right) \sqrt{3}-72 i \sqrt{26}+181\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}}{25392}-\frac{i \sqrt{3}\, \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{11}{24}\right)^{-n}}{1760512}\\+\\\frac{\left(\left(\left(4901 i+720 \sqrt{26}\right) \sqrt{3}-2160 i \sqrt{26}-4901\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}+233818+\left(\left(-20033 i+552 \sqrt{26}\right) \sqrt{3}+1656 i \sqrt{26}-20033\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-181 i-24 \sqrt{26}\right) \sqrt{3}+72 i \sqrt{26}+181\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}}{25392}+\frac{i \sqrt{3}\, \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{11}{24}\right)^{-n}}{1760512}\\-\\\frac{3 \left(\left(\frac{30 \sqrt{26}\, \sqrt{3}}{23}-\frac{4901}{552}\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}+\sqrt{3}\, \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{26}-\frac{871 \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}-\frac{5083}{24}\right) \left(-\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{11}{24}-\frac{181 \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}}{12696}+\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{26}\, \sqrt{3}}{529}\right)^{-n}}{4784} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 36 rules.
Finding the specification took 42 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_{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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{2}\! \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_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{31}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{30}\! \left(x \right)\\
\end{align*}\)