Av(12453, 12543, 21453, 21543, 31452, 31542)
Counting Sequence
1, 1, 2, 6, 24, 114, 596, 3294, 18852, 110486, 658808, 3980626, 24305912, 149698890, 928690892, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}-x^{2}-x +1\right) F \left(x
\right)^{4}+\left(-x^{3}-2 x^{2}+6 x -5\right) F \left(x
\right)^{3}+\left(4 x^{2}-9 x +9\right) F \left(x
\right)^{2}+\left(-x^{2}+4 x -7\right) F \! \left(x \right)+2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 114\)
\(\displaystyle a \! \left(6\right) = 596\)
\(\displaystyle a \! \left(7\right) = 3294\)
\(\displaystyle a \! \left(8\right) = 18852\)
\(\displaystyle a \! \left(9\right) = 110486\)
\(\displaystyle a \! \left(10\right) = 658808\)
\(\displaystyle a \! \left(11\right) = 3980626\)
\(\displaystyle a \! \left(12\right) = 24305912\)
\(\displaystyle a \! \left(13\right) = 149698890\)
\(\displaystyle a \! \left(14\right) = 928690892\)
\(\displaystyle a \! \left(15\right) = 5797176310\)
\(\displaystyle a \! \left(16\right) = 36383341132\)
\(\displaystyle a \! \left(17\right) = 229429650654\)
\(\displaystyle a \! \left(n +18\right) = -\frac{4 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{8 \left(n +1\right) \left(17 n^{2}+64 n +57\right) a \! \left(n +1\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(7922 n^{3}+61755 n^{2}+156199 n +128448\right) a \! \left(n +2\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(26436 n^{3}+284669 n^{2}+1005886 n +1167452\right) a \! \left(n +3\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(451702 n^{3}+6211290 n^{2}+28157234 n +42123609\right) a \! \left(n +4\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(493794 n^{3}+8284088 n^{2}+45873865 n +84001439\right) a \! \left(n +5\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(855090 n^{3}+17002472 n^{2}+111165709 n +239926235\right) a \! \left(n +6\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(1353200 n^{3}+32286693 n^{2}+243072904 n +587342763\right) a \! \left(n +7\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(741902 n^{3}+17085385 n^{2}+138338296 n +390297247\right) a \! \left(n +8\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(892386 n^{3}+21433717 n^{2}+175407309 n +488519437\right) a \! \left(n +9\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(271670 n^{3}+3484188 n^{2}-1521935 n -99617496\right) a \! \left(n +10\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(271850 n^{3}+8121669 n^{2}+80637130 n +265838784\right) a \! \left(n +11\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(253120 n^{3}+7442178 n^{2}+71530322 n +223027245\right) a \! \left(n +12\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(24148 n^{3}+1068744 n^{2}+15153587 n +69645555\right) a \! \left(n +13\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(14240 n^{3}+538086 n^{2}+6710851 n +27579705\right) a \! \left(n +14\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(362 n^{3}+19015 n^{2}+323488 n +1795483\right) a \! \left(n +15\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(316 n^{3}+14705 n^{2}+228064 n +1178845\right) a \! \left(n +16\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(44 n^{3}+2125 n^{2}+34189 n +183255\right) a \! \left(n +17\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}, \quad n \geq 18\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 114\)
\(\displaystyle a \! \left(6\right) = 596\)
\(\displaystyle a \! \left(7\right) = 3294\)
\(\displaystyle a \! \left(8\right) = 18852\)
\(\displaystyle a \! \left(9\right) = 110486\)
\(\displaystyle a \! \left(10\right) = 658808\)
\(\displaystyle a \! \left(11\right) = 3980626\)
\(\displaystyle a \! \left(12\right) = 24305912\)
\(\displaystyle a \! \left(13\right) = 149698890\)
\(\displaystyle a \! \left(14\right) = 928690892\)
\(\displaystyle a \! \left(15\right) = 5797176310\)
\(\displaystyle a \! \left(16\right) = 36383341132\)
\(\displaystyle a \! \left(17\right) = 229429650654\)
\(\displaystyle a \! \left(n +18\right) = -\frac{4 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{8 \left(n +1\right) \left(17 n^{2}+64 n +57\right) a \! \left(n +1\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(7922 n^{3}+61755 n^{2}+156199 n +128448\right) a \! \left(n +2\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(26436 n^{3}+284669 n^{2}+1005886 n +1167452\right) a \! \left(n +3\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(451702 n^{3}+6211290 n^{2}+28157234 n +42123609\right) a \! \left(n +4\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(493794 n^{3}+8284088 n^{2}+45873865 n +84001439\right) a \! \left(n +5\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(855090 n^{3}+17002472 n^{2}+111165709 n +239926235\right) a \! \left(n +6\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(1353200 n^{3}+32286693 n^{2}+243072904 n +587342763\right) a \! \left(n +7\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(741902 n^{3}+17085385 n^{2}+138338296 n +390297247\right) a \! \left(n +8\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(892386 n^{3}+21433717 n^{2}+175407309 n +488519437\right) a \! \left(n +9\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(271670 n^{3}+3484188 n^{2}-1521935 n -99617496\right) a \! \left(n +10\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(271850 n^{3}+8121669 n^{2}+80637130 n +265838784\right) a \! \left(n +11\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(253120 n^{3}+7442178 n^{2}+71530322 n +223027245\right) a \! \left(n +12\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(24148 n^{3}+1068744 n^{2}+15153587 n +69645555\right) a \! \left(n +13\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(14240 n^{3}+538086 n^{2}+6710851 n +27579705\right) a \! \left(n +14\right)}{3 \left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(362 n^{3}+19015 n^{2}+323488 n +1795483\right) a \! \left(n +15\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}-\frac{\left(316 n^{3}+14705 n^{2}+228064 n +1178845\right) a \! \left(n +16\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}+\frac{\left(44 n^{3}+2125 n^{2}+34189 n +183255\right) a \! \left(n +17\right)}{\left(2 n +33\right) \left(n +18\right) \left(n +16\right)}, \quad n \geq 18\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 36 rules.
Found on January 23, 2022.Finding the specification took 75 seconds.
Copy 36 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_{17}\! \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 \right)\\
F_{6}\! \left(x \right) &= F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{17}\! \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_{24}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{17}\! \left(x \right) F_{29}\! \left(x \right) F_{32}\! \left(x \right)\\
\end{align*}\)