Av(13452, 13542, 14352, 31452, 31542)
Counting Sequence
1, 1, 2, 6, 24, 115, 615, 3526, 21189, 131695, 839493, 5457909, 36049457, 241212167, 1631550587, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-x +1\right) F \left(x
\right)^{4}+\left(-x^{2}+3 x -4\right) F \left(x
\right)^{3}+\left(x^{2}-3 x +6\right) F \left(x
\right)^{2}+\left(x -4\right) F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 115\)
\(\displaystyle a \! \left(6\right) = 615\)
\(\displaystyle a \! \left(7\right) = 3526\)
\(\displaystyle a \! \left(8\right) = 21189\)
\(\displaystyle a \! \left(9\right) = 131695\)
\(\displaystyle a \! \left(10\right) = 839493\)
\(\displaystyle a \! \left(11\right) = 5457909\)
\(\displaystyle a \! \left(12\right) = 36049457\)
\(\displaystyle a \! \left(n +13\right) = \frac{21 \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{\left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(n +2\right) \left(2032 n^{2}+9181 n +9105\right) a \! \left(n +1\right)}{3 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{2 \left(21533 n^{3}+202824 n^{2}+621154 n +620787\right) a \! \left(n +2\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(36501 n^{3}+457060 n^{2}+1866067 n +2496804\right) a \! \left(n +3\right)}{2 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(711623 n^{3}+11792234 n^{2}+62925435 n +109452768\right) a \! \left(n +4\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(388612 n^{3}+9114363 n^{2}+64449851 n +144303840\right) a \! \left(n +5\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(81116 n^{3}+6193479 n^{2}+66902801 n +198124938\right) a \! \left(n +6\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(1957259 n^{3}+34807718 n^{2}+207880287 n +418555764\right) a \! \left(n +7\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(2592129 n^{3}+54685088 n^{2}+386580871 n +917727144\right) a \! \left(n +8\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(1382477 n^{3}+30678056 n^{2}+225553835 n +549383232\right) a \! \left(n +9\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(4331 n^{3}+352140 n^{2}+5256837 n +21911508\right) a \! \left(n +10\right)}{6 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{4 \left(3385 n^{3}+103002 n^{2}+1043715 n +3521684\right) a \! \left(n +11\right)}{3 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(513 n^{3}+16632 n^{2}+179579 n +645716\right) a \! \left(n +12\right)}{\left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 115\)
\(\displaystyle a \! \left(6\right) = 615\)
\(\displaystyle a \! \left(7\right) = 3526\)
\(\displaystyle a \! \left(8\right) = 21189\)
\(\displaystyle a \! \left(9\right) = 131695\)
\(\displaystyle a \! \left(10\right) = 839493\)
\(\displaystyle a \! \left(11\right) = 5457909\)
\(\displaystyle a \! \left(12\right) = 36049457\)
\(\displaystyle a \! \left(n +13\right) = \frac{21 \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{\left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(n +2\right) \left(2032 n^{2}+9181 n +9105\right) a \! \left(n +1\right)}{3 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{2 \left(21533 n^{3}+202824 n^{2}+621154 n +620787\right) a \! \left(n +2\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(36501 n^{3}+457060 n^{2}+1866067 n +2496804\right) a \! \left(n +3\right)}{2 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(711623 n^{3}+11792234 n^{2}+62925435 n +109452768\right) a \! \left(n +4\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(388612 n^{3}+9114363 n^{2}+64449851 n +144303840\right) a \! \left(n +5\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(81116 n^{3}+6193479 n^{2}+66902801 n +198124938\right) a \! \left(n +6\right)}{9 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(1957259 n^{3}+34807718 n^{2}+207880287 n +418555764\right) a \! \left(n +7\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{\left(2592129 n^{3}+54685088 n^{2}+386580871 n +917727144\right) a \! \left(n +8\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(1382477 n^{3}+30678056 n^{2}+225553835 n +549383232\right) a \! \left(n +9\right)}{18 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(4331 n^{3}+352140 n^{2}+5256837 n +21911508\right) a \! \left(n +10\right)}{6 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}-\frac{4 \left(3385 n^{3}+103002 n^{2}+1043715 n +3521684\right) a \! \left(n +11\right)}{3 \left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}+\frac{\left(513 n^{3}+16632 n^{2}+179579 n +645716\right) a \! \left(n +12\right)}{\left(3 n +35\right) \left(3 n +37\right) \left(n +12\right)}, \quad n \geq 13\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 24 rules.
Found on January 23, 2022.Finding the specification took 200 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)^{2} F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= y x\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{23}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{20}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\
F_{23}\! \left(x \right) &= x\\
\end{align*}\)