Av(13452, 13542, 15342, 31452, 31542)
Counting Sequence
1, 1, 2, 6, 24, 115, 614, 3505, 20910, 128672, 810230, 5193634, 33769514, 222155467, 1475842608, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(7 x^{2}-8 x +4\right) F \left(x
\right)^{4}+\left(-10 x^{2}+17 x -14\right) F \left(x
\right)^{3}+\left(x^{2}-7 x +17\right) F \left(x
\right)^{2}+\left(-2 x -8\right) F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 115\)
\(\displaystyle a(6) = 614\)
\(\displaystyle a(7) = 3505\)
\(\displaystyle a(8) = 20910\)
\(\displaystyle a(9) = 128672\)
\(\displaystyle a(10) = 810230\)
\(\displaystyle a(11) = 5193634\)
\(\displaystyle a(12) = 33769514\)
\(\displaystyle a(13) = 222155467\)
\(\displaystyle a(14) = 1475842608\)
\(\displaystyle a{\left(n + 15 \right)} = \frac{23520 n \left(n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{13 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{84 \left(n + 1\right) \left(921142 n^{2} + 3047361 n + 2603825\right) a{\left(n + 1 \right)}}{221 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(119941 n + 1496151\right) a{\left(n + 14 \right)}}{4420 \left(n + 15\right)} - \frac{3 \left(98055 n^{2} + 2363339 n + 14243295\right) a{\left(n + 13 \right)}}{1105 \left(n + 14\right) \left(n + 15\right)} + \frac{3 \left(5202895 n^{3} + 184707754 n^{2} + 2184056377 n + 8601403414\right) a{\left(n + 12 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(45248195 n^{3} + 1333695150 n^{2} + 13024486813 n + 42111490314\right) a{\left(n + 11 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(84516265 n^{3} + 8805943680 n^{2} + 135654615353 n + 568761152550\right) a{\left(n + 9 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{2 \left(465045284 n^{3} + 3520636908 n^{2} + 8848240087 n + 7332570777\right) a{\left(n + 2 \right)}}{221 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(815620633 n^{3} + 24522535518 n^{2} + 245794151987 n + 821150134014\right) a{\left(n + 10 \right)}}{35360 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(1578512416 n^{3} - 29318002758 n^{2} - 236597657533 n - 403849823145\right) a{\left(n + 3 \right)}}{2210 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(8835908798 n^{3} + 187982014959 n^{2} + 1317612532057 n + 3042173271366\right) a{\left(n + 7 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(10468607327 n^{3} + 202587412662 n^{2} + 1240034918893 n + 2317011782094\right) a{\left(n + 8 \right)}}{35360 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(15880113314 n^{3} + 211915983987 n^{2} + 834743230885 n + 817295554836\right) a{\left(n + 6 \right)}}{8840 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(26010513056 n^{3} + 409904419278 n^{2} + 1957387191721 n + 2946391624575\right) a{\left(n + 4 \right)}}{4420 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{3 \left(38423869170 n^{3} + 540657959179 n^{2} + 2462404222323 n + 3613028734150\right) a{\left(n + 5 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)}, \quad n \geq 15\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 115\)
\(\displaystyle a(6) = 614\)
\(\displaystyle a(7) = 3505\)
\(\displaystyle a(8) = 20910\)
\(\displaystyle a(9) = 128672\)
\(\displaystyle a(10) = 810230\)
\(\displaystyle a(11) = 5193634\)
\(\displaystyle a(12) = 33769514\)
\(\displaystyle a(13) = 222155467\)
\(\displaystyle a(14) = 1475842608\)
\(\displaystyle a{\left(n + 15 \right)} = \frac{23520 n \left(n + 1\right) \left(2 n + 3\right) a{\left(n \right)}}{13 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{84 \left(n + 1\right) \left(921142 n^{2} + 3047361 n + 2603825\right) a{\left(n + 1 \right)}}{221 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(119941 n + 1496151\right) a{\left(n + 14 \right)}}{4420 \left(n + 15\right)} - \frac{3 \left(98055 n^{2} + 2363339 n + 14243295\right) a{\left(n + 13 \right)}}{1105 \left(n + 14\right) \left(n + 15\right)} + \frac{3 \left(5202895 n^{3} + 184707754 n^{2} + 2184056377 n + 8601403414\right) a{\left(n + 12 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(45248195 n^{3} + 1333695150 n^{2} + 13024486813 n + 42111490314\right) a{\left(n + 11 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(84516265 n^{3} + 8805943680 n^{2} + 135654615353 n + 568761152550\right) a{\left(n + 9 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{2 \left(465045284 n^{3} + 3520636908 n^{2} + 8848240087 n + 7332570777\right) a{\left(n + 2 \right)}}{221 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(815620633 n^{3} + 24522535518 n^{2} + 245794151987 n + 821150134014\right) a{\left(n + 10 \right)}}{35360 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(1578512416 n^{3} - 29318002758 n^{2} - 236597657533 n - 403849823145\right) a{\left(n + 3 \right)}}{2210 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(8835908798 n^{3} + 187982014959 n^{2} + 1317612532057 n + 3042173271366\right) a{\left(n + 7 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{\left(10468607327 n^{3} + 202587412662 n^{2} + 1240034918893 n + 2317011782094\right) a{\left(n + 8 \right)}}{35360 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(15880113314 n^{3} + 211915983987 n^{2} + 834743230885 n + 817295554836\right) a{\left(n + 6 \right)}}{8840 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} - \frac{\left(26010513056 n^{3} + 409904419278 n^{2} + 1957387191721 n + 2946391624575\right) a{\left(n + 4 \right)}}{4420 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)} + \frac{3 \left(38423869170 n^{3} + 540657959179 n^{2} + 2462404222323 n + 3613028734150\right) a{\left(n + 5 \right)}}{17680 \left(n + 13\right) \left(n + 14\right) \left(n + 15\right)}, \quad n \geq 15\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 21 rules.
Finding the specification took 158 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+4 x^{2} F_{7}\! \left(x , y\right) y^{2}+4 y^{2} x^{2}+4 x F_{7}\! \left(x , y\right)^{2} y -5 x F_{7}\! \left(x , y\right) y -y x -F_{7}\! \left(x , y\right)^{2}+2 F_{7}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{20}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\
F_{17}\! \left(x , y\right) &= y x\\
F_{19}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\
F_{20}\! \left(x \right) &= x\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 27 rules.
Finding the specification took 117 seconds.
Copy 27 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_{26}\! \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_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+4 x^{2} F_{7}\! \left(x , y\right) y^{2}+4 y^{2} x^{2}+4 x F_{7}\! \left(x , y\right)^{2} y -5 x F_{7}\! \left(x , y\right) y -y x -F_{7}\! \left(x , y\right)^{2}+2 F_{7}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{26}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\
F_{17}\! \left(x , y\right) &= y x\\
F_{19}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{18}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{2}\! \left(x \right)\\
F_{26}\! \left(x \right) &= x\\
\end{align*}\)