Av(1234, 1243, 2314, 3241, 4213)
Generating Function
\(\displaystyle -\frac{x^{14}+3 x^{13}-7 x^{12}-27 x^{11}-2 x^{10}+43 x^{9}+14 x^{8}-24 x^{7}-19 x^{6}+2 x^{5}+19 x^{4}-x^{3}-11 x^{2}+6 x -1}{\left(x^{2}+2 x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 118, 271, 619, 1411, 3223, 7390, 17033, 39490, 92111, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+2 x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{3} F \! \left(x \right)+x^{14}+3 x^{13}-7 x^{12}-27 x^{11}-2 x^{10}+43 x^{9}+14 x^{8}-24 x^{7}-19 x^{6}+2 x^{5}+19 x^{4}-x^{3}-11 x^{2}+6 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 118\)
\(\displaystyle a \! \left(7\right) = 271\)
\(\displaystyle a \! \left(8\right) = 619\)
\(\displaystyle a \! \left(9\right) = 1411\)
\(\displaystyle a \! \left(10\right) = 3223\)
\(\displaystyle a \! \left(11\right) = 7390\)
\(\displaystyle a \! \left(12\right) = 17033\)
\(\displaystyle a \! \left(13\right) = 39490\)
\(\displaystyle a \! \left(14\right) = 92111\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-5 a \! \left(n +1\right)-5 a \! \left(n +2\right)+8 a \! \left(n +3\right)+10 a \! \left(n +4\right)-8 a \! \left(n +5\right)-5 a \! \left(n +6\right)+5 a \! \left(n +7\right)+4 n -28, \quad n \geq 15\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 118\)
\(\displaystyle a \! \left(7\right) = 271\)
\(\displaystyle a \! \left(8\right) = 619\)
\(\displaystyle a \! \left(9\right) = 1411\)
\(\displaystyle a \! \left(10\right) = 3223\)
\(\displaystyle a \! \left(11\right) = 7390\)
\(\displaystyle a \! \left(12\right) = 17033\)
\(\displaystyle a \! \left(13\right) = 39490\)
\(\displaystyle a \! \left(14\right) = 92111\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-5 a \! \left(n +1\right)-5 a \! \left(n +2\right)+8 a \! \left(n +3\right)+10 a \! \left(n +4\right)-8 a \! \left(n +5\right)-5 a \! \left(n +6\right)+5 a \! \left(n +7\right)+4 n -28, \quad n \geq 15\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 19 & n =4 \\ \frac{\left(\left(-5 n^{2}+55 n -178\right) \sqrt{5}+25 n^{2}+85 n -250\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{500}+\\\frac{\left(\left(5 n^{2}-55 n +178\right) \sqrt{5}+25 n^{2}+85 n -250\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{500}-\\\frac{\left(-1-\sqrt{2}\right)^{-n} \sqrt{2}}{4}+\frac{\left(\sqrt{2}-1\right)^{-n} \sqrt{2}}{4}+2 n -8 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 75 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 75 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{24}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{25}\! \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_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{40}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{73}\! \left(x \right)\\
\end{align*}\)