Av(12453, 12534, 12543, 13452, 13542, 14352, 21453, 21534, 21543, 23451, 23541, 24351, 31452, 31524, 31542, 32451, 32541, 34251, 41352, 41523, 41532, 42351, 42531, 43251, 51342, 51423, 51432, 52341, 52431, 53241)
Generating Function
\(\displaystyle \frac{4 x^{7}+12 x^{6}+22 x^{5}+16 x^{4}+2 x^{3}-2 x^{2}-3 x +1}{\left(2 x^{3}+2 x^{2}+2 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 312, 1064, 3552, 11664, 37840, 121528, 387072, 1224288, 3849440, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x^{3}+2 x^{2}+2 x -1\right)^{2} F \! \left(x \right)+4 x^{7}+12 x^{6}+22 x^{5}+16 x^{4}+2 x^{3}-2 x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 312\)
\(\displaystyle a(7) = 1064\)
\(\displaystyle a{\left(n + 3 \right)} = - a{\left(n \right)} - 2 a{\left(n + 1 \right)} - 3 a{\left(n + 2 \right)} + a{\left(n + 5 \right)} - \frac{a{\left(n + 6 \right)}}{4}, \quad n \geq 8\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 312\)
\(\displaystyle a(7) = 1064\)
\(\displaystyle a{\left(n + 3 \right)} = - a{\left(n \right)} - 2 a{\left(n + 1 \right)} - 3 a{\left(n + 2 \right)} + a{\left(n + 5 \right)} - \frac{a{\left(n + 6 \right)}}{4}, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ -\frac{2^{\frac{1}{3}} \left(\left(2^{\frac{1}{3}} \left(\left(i+\frac{19 \sqrt{67}}{201}\right) \sqrt{3}-\frac{19 i \sqrt{67}}{67}-1\right) \left(41+3 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-4 i+\frac{44 \sqrt{67}}{201}\right) \sqrt{3}+\frac{44 i \sqrt{67}}{67}-4\right) \left(\frac{41 \left(\left(i+\frac{3 \sqrt{67}}{41}\right) \sqrt{3}-\frac{9 i \sqrt{67}}{41}-1\right) 2^{\frac{2}{3}} \left(41+3 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{i \sqrt{3}\, \left(82+6 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(82+6 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}+\left(-\left(\left(i-\frac{19 \sqrt{67}}{201}\right) \sqrt{3}-\frac{19 i \sqrt{67}}{67}+1\right) 2^{\frac{1}{3}} \left(41+3 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(4 i+\frac{44 \sqrt{67}}{201}\right) \sqrt{3}-\frac{44 i \sqrt{67}}{67}-4\right) \left(-\frac{41 \left(\left(i-\frac{3 \sqrt{67}}{41}\right) \sqrt{3}-\frac{9 i \sqrt{67}}{41}+1\right) 2^{\frac{2}{3}} \left(41+3 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{i \sqrt{3}\, \left(82+6 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(82+6 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}-\frac{38 \left(2^{\frac{1}{3}} \left(\sqrt{67}\, \sqrt{3}-\frac{201}{19}\right) \left(41+3 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{44 \sqrt{67}\, \sqrt{3}}{19}-\frac{804}{19}\right) \left(\frac{\left(-3 \sqrt{67}\, \sqrt{3}+41\right) 2^{\frac{2}{3}} \left(41+3 \sqrt{67}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}+\frac{\left(82+6 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{201}\right) \left(41+3 \sqrt{67}\, \sqrt{3}\right)^{\frac{1}{3}} n}{128} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 76 rules.
Finding the specification took 91 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 76 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_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \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_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{11}\! \left(x \right) &= 0\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= F_{19}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{19}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{19}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{19}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{19}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{19}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 2 F_{11}\! \left(x \right)+F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{19}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{19}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{11}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{19}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{19}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{61}\! \left(x \right) &= 0\\
F_{62}\! \left(x \right) &= F_{19}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{19}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{19}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{19}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{10}\! \left(x \right)\\
\end{align*}\)