Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432, 23154, 23451, 23541, 24153, 24351, 24531, 25143, 25341, 25431, 34152, 34251, 35142, 35241)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(3 x^{4}-6 x^{3}+8 x^{2}-5 x +1\right)}{\left(8 x^{3}-11 x^{2}+6 x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 95, 354, 1270, 4484, 15763, 55410, 194934, 686192, 2416151, 8508258, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(8 x^{3}-11 x^{2}+6 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-\left(2 x -1\right) \left(3 x^{4}-6 x^{3}+8 x^{2}-5 x +1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a{\left(n + 3 \right)} = - n + 8 a{\left(n \right)} - 11 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)} + 3, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a{\left(n + 3 \right)} = - n + 8 a{\left(n \right)} - 11 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)} + 3, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(759 \sqrt{26}+7774 i\right) \sqrt{3}-2277 i \sqrt{26}-7774\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}+137540+\left(\left(93 \sqrt{26}+26 i\right) \sqrt{3}+279 i \sqrt{26}+26\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(181 i-24 \sqrt{26}\right) \sqrt{3}-72 i \sqrt{26}+181\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}}{25392}-\frac{i \sqrt{3}\, \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{11}{24}\right)^{-n}}{330096}\\+\\\frac{\left(\left(\left(93 \sqrt{26}-26 i\right) \sqrt{3}-279 i \sqrt{26}+26\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}+137540+\left(\left(759 \sqrt{26}-7774 i\right) \sqrt{3}+2277 i \sqrt{26}-7774\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-181 i-24 \sqrt{26}\right) \sqrt{3}+72 i \sqrt{26}+181\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}}{25392}+\frac{i \sqrt{3}\, \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{11}{24}\right)^{-n}}{330096}\\+\\\frac{\left(\left(-186 \sqrt{26}\, \sqrt{3}-52\right) \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}-1518 \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{26}\, \sqrt{3}+15548 \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}+137540\right) \left(-\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{11}{24}-\frac{181 \left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}}}{12696}+\frac{\left(181+24 \sqrt{26}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{26}\, \sqrt{3}}{529}\right)^{-n}}{330096}\\+\frac{n}{2}-1 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 45 rules.
Finding the specification took 279 seconds.
Copy 45 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_{16}\! \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_{27}\! \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_{16}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \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_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{41}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{24}\! \left(x \right)\\
\end{align*}\)