Av(1243, 2413, 3142)
Counting Sequence
1, 1, 2, 6, 21, 77, 288, 1093, 4202, 16341, 64187, 254313, 1015163, 4078777, 16481961, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} x^{2} F \left(x
\right)^{4}+x \left(x -1\right) \left(x +1\right) F \left(x
\right)^{3}+x \left(3 x^{2}-6 x +4\right) F \left(x
\right)^{2}+\left(x -1\right) \left(x +1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 288\)
\(\displaystyle a \! \left(7\right) = 1093\)
\(\displaystyle a \! \left(8\right) = 4202\)
\(\displaystyle a \! \left(9\right) = 16341\)
\(\displaystyle a \! \left(10\right) = 64187\)
\(\displaystyle a \! \left(11\right) = 254313\)
\(\displaystyle a \! \left(12\right) = 1015163\)
\(\displaystyle a \! \left(13\right) = 4078777\)
\(\displaystyle a \! \left(14\right) = 16481961\)
\(\displaystyle a \! \left(15\right) = 66940960\)
\(\displaystyle a \! \left(n +16\right) = -\frac{50 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{3 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{2 \left(2 n +3\right) \left(441 n^{2}+2173 n +2630\right) a \! \left(n +1\right)}{\left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{\left(138487 n^{3}+1266412 n^{2}+3813521 n +3782340\right) a \! \left(n +2\right)}{10 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(575855 n^{3}+6800348 n^{2}+26642401 n +34632300\right) a \! \left(n +3\right)}{10 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(1163144 n^{3}+16935009 n^{2}+82085491 n +132442818\right) a \! \left(n +4\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(4504681 n^{3}+78536592 n^{2}+456581717 n +884978454\right) a \! \left(n +5\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(3326038 n^{3}+67838094 n^{2}+461606633 n +1047771843\right) a \! \left(n +6\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(7695121 n^{3}+179780781 n^{2}+1401060188 n +3641914452\right) a \! \left(n +7\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{\left(13999109 n^{3}+367691094 n^{2}+3220260577 n +9404266560\right) a \! \left(n +8\right)}{30 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(4958239 n^{3}+144173976 n^{2}+1397347133 n +4514361996\right) a \! \left(n +9\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(1346029 n^{3}+42817533 n^{2}+453818030 n +1602708864\right) a \! \left(n +10\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(2196607 n^{3}+75753144 n^{2}+870091997 n +3328571580\right) a \! \left(n +11\right)}{30 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(54691 n^{3}+2030518 n^{2}+25096826 n +103263207\right) a \! \left(n +12\right)}{5 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(69182 n^{3}+2749671 n^{2}+36364411 n +160009842\right) a \! \left(n +13\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(4846 n^{3}+205215 n^{2}+2890118 n +13534179\right) a \! \left(n +14\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{2 \left(403 n^{3}+18108 n^{2}+270443 n +1342158\right) a \! \left(n +15\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}, \quad n \geq 16\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 288\)
\(\displaystyle a \! \left(7\right) = 1093\)
\(\displaystyle a \! \left(8\right) = 4202\)
\(\displaystyle a \! \left(9\right) = 16341\)
\(\displaystyle a \! \left(10\right) = 64187\)
\(\displaystyle a \! \left(11\right) = 254313\)
\(\displaystyle a \! \left(12\right) = 1015163\)
\(\displaystyle a \! \left(13\right) = 4078777\)
\(\displaystyle a \! \left(14\right) = 16481961\)
\(\displaystyle a \! \left(15\right) = 66940960\)
\(\displaystyle a \! \left(n +16\right) = -\frac{50 \left(2 n +3\right) \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{3 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{2 \left(2 n +3\right) \left(441 n^{2}+2173 n +2630\right) a \! \left(n +1\right)}{\left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{\left(138487 n^{3}+1266412 n^{2}+3813521 n +3782340\right) a \! \left(n +2\right)}{10 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(575855 n^{3}+6800348 n^{2}+26642401 n +34632300\right) a \! \left(n +3\right)}{10 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(1163144 n^{3}+16935009 n^{2}+82085491 n +132442818\right) a \! \left(n +4\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(4504681 n^{3}+78536592 n^{2}+456581717 n +884978454\right) a \! \left(n +5\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(3326038 n^{3}+67838094 n^{2}+461606633 n +1047771843\right) a \! \left(n +6\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(7695121 n^{3}+179780781 n^{2}+1401060188 n +3641914452\right) a \! \left(n +7\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{\left(13999109 n^{3}+367691094 n^{2}+3220260577 n +9404266560\right) a \! \left(n +8\right)}{30 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(4958239 n^{3}+144173976 n^{2}+1397347133 n +4514361996\right) a \! \left(n +9\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(1346029 n^{3}+42817533 n^{2}+453818030 n +1602708864\right) a \! \left(n +10\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(2196607 n^{3}+75753144 n^{2}+870091997 n +3328571580\right) a \! \left(n +11\right)}{30 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(54691 n^{3}+2030518 n^{2}+25096826 n +103263207\right) a \! \left(n +12\right)}{5 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{\left(69182 n^{3}+2749671 n^{2}+36364411 n +160009842\right) a \! \left(n +13\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}-\frac{2 \left(4846 n^{3}+205215 n^{2}+2890118 n +13534179\right) a \! \left(n +14\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}+\frac{2 \left(403 n^{3}+18108 n^{2}+270443 n +1342158\right) a \! \left(n +15\right)}{15 \left(2 n +33\right) \left(n +17\right) \left(n +14\right)}, \quad n \geq 16\)
This specification was found using the strategy pack "Point Placements" and has 16 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 16 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
\end{align*}\)