Av(1234, 1243, 1324)
Counting Sequence
1, 1, 2, 6, 21, 80, 322, 1346, 5783, 25372, 113174, 511649, 2338988, 10793251, 50205607, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{2}+8 x -1\right) F \left(x
\right)^{4}+\left(x -5\right) \left(x^{2}+9 x -1\right) F \left(x
\right)^{3}+\left(3 x^{3}-21 x^{2}+94 x -9\right) F \left(x
\right)^{2}+\left(x^{3}+12 x^{2}-82 x +7\right) F \! \left(x \right)+3 x^{2}+26 x -2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 80\)
\(\displaystyle a \! \left(6\right) = 322\)
\(\displaystyle a \! \left(7\right) = 1346\)
\(\displaystyle a \! \left(8\right) = 5783\)
\(\displaystyle a \! \left(9\right) = 25372\)
\(\displaystyle a \! \left(10\right) = 113174\)
\(\displaystyle a \! \left(11\right) = 511649\)
\(\displaystyle a \! \left(12\right) = 2338988\)
\(\displaystyle a \! \left(13\right) = 10793251\)
\(\displaystyle a \! \left(14\right) = 50205607\)
\(\displaystyle a \! \left(15\right) = 235156609\)
\(\displaystyle a \! \left(16\right) = 1108120540\)
\(\displaystyle a \! \left(n +17\right) = \frac{64 \left(n -1\right) n \left(n +1\right) a \! \left(n \right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{8 n \left(163 n +281\right) \left(n +1\right) a \! \left(n +1\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{16 \left(n +1\right) \left(37 n^{2}+596 n +6\right) a \! \left(n +2\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(5473 n^{3}+38232 n^{2}+101151 n +98960\right) a \! \left(n +3\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(8421 n^{3}+73066 n^{2}+197441 n +156436\right) a \! \left(n +4\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{6 \left(30245 n^{3}+368164 n^{2}+1510787 n +2103516\right) a \! \left(n +5\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(7030 n^{3}+65845 n^{2}+121061 n -160182\right) a \! \left(n +6\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{3 \left(18065 n^{3}+226928 n^{2}+833127 n +778152\right) a \! \left(n +7\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{3 \left(1549 n^{3}+24230 n^{2}+180289 n +636096\right) a \! \left(n +8\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(23413 n^{3}+652944 n^{2}+6074141 n +18810402\right) a \! \left(n +9\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{2 \left(20173 n^{3}+619098 n^{2}+6299948 n +21273837\right) a \! \left(n +10\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(23947 n^{3}+758244 n^{2}+7955477 n +27658776\right) a \! \left(n +11\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{\left(12937 n^{3}+436518 n^{2}+4887083 n +18148686\right) a \! \left(n +12\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(2447 n^{3}+84444 n^{2}+958105 n +3561192\right) a \! \left(n +13\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{2 \left(19 n^{3}+1200 n^{2}+21932 n +124167\right) a \! \left(n +14\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(n^{2}+237 n +3302\right) a \! \left(n +15\right)}{\left(n +17\right) \left(n +16\right)}+\frac{\left(11 n +193\right) a \! \left(n +16\right)}{n +17}, \quad n \geq 17\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 80\)
\(\displaystyle a \! \left(6\right) = 322\)
\(\displaystyle a \! \left(7\right) = 1346\)
\(\displaystyle a \! \left(8\right) = 5783\)
\(\displaystyle a \! \left(9\right) = 25372\)
\(\displaystyle a \! \left(10\right) = 113174\)
\(\displaystyle a \! \left(11\right) = 511649\)
\(\displaystyle a \! \left(12\right) = 2338988\)
\(\displaystyle a \! \left(13\right) = 10793251\)
\(\displaystyle a \! \left(14\right) = 50205607\)
\(\displaystyle a \! \left(15\right) = 235156609\)
\(\displaystyle a \! \left(16\right) = 1108120540\)
\(\displaystyle a \! \left(n +17\right) = \frac{64 \left(n -1\right) n \left(n +1\right) a \! \left(n \right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{8 n \left(163 n +281\right) \left(n +1\right) a \! \left(n +1\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{16 \left(n +1\right) \left(37 n^{2}+596 n +6\right) a \! \left(n +2\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(5473 n^{3}+38232 n^{2}+101151 n +98960\right) a \! \left(n +3\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(8421 n^{3}+73066 n^{2}+197441 n +156436\right) a \! \left(n +4\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{6 \left(30245 n^{3}+368164 n^{2}+1510787 n +2103516\right) a \! \left(n +5\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{12 \left(7030 n^{3}+65845 n^{2}+121061 n -160182\right) a \! \left(n +6\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{3 \left(18065 n^{3}+226928 n^{2}+833127 n +778152\right) a \! \left(n +7\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{3 \left(1549 n^{3}+24230 n^{2}+180289 n +636096\right) a \! \left(n +8\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(23413 n^{3}+652944 n^{2}+6074141 n +18810402\right) a \! \left(n +9\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{2 \left(20173 n^{3}+619098 n^{2}+6299948 n +21273837\right) a \! \left(n +10\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(23947 n^{3}+758244 n^{2}+7955477 n +27658776\right) a \! \left(n +11\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}+\frac{\left(12937 n^{3}+436518 n^{2}+4887083 n +18148686\right) a \! \left(n +12\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(2447 n^{3}+84444 n^{2}+958105 n +3561192\right) a \! \left(n +13\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{2 \left(19 n^{3}+1200 n^{2}+21932 n +124167\right) a \! \left(n +14\right)}{\left(n +15\right) \left(n +17\right) \left(n +16\right)}-\frac{\left(n^{2}+237 n +3302\right) a \! \left(n +15\right)}{\left(n +17\right) \left(n +16\right)}+\frac{\left(11 n +193\right) a \! \left(n +16\right)}{n +17}, \quad n \geq 17\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 18 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 18 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{15}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{15}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{7}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{6}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= y x\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{15}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\
F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{15}\! \left(x \right)\\
F_{14}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x , 1\right)\\
\end{align*}\)