Av(1243, 2143, 2413, 3142)
Counting Sequence
1, 1, 2, 6, 20, 70, 253, 936, 3525, 13463, 52008, 202814, 797236, 3155323, 12562600, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} F \left(x
\right)^{4}-\left(3 x +1\right) x F \left(x
\right)^{3}+x \left(x^{2}+5\right) F \left(x
\right)^{2}+\left(-3 x -1\right) F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 70\)
\(\displaystyle a \! \left(6\right) = 253\)
\(\displaystyle a \! \left(7\right) = 936\)
\(\displaystyle a \! \left(8\right) = 3525\)
\(\displaystyle a \! \left(9\right) = 13463\)
\(\displaystyle a \! \left(10\right) = 52008\)
\(\displaystyle a \! \left(11\right) = 202814\)
\(\displaystyle a \! \left(12\right) = 797236\)
\(\displaystyle a \! \left(13\right) = 3155323\)
\(\displaystyle a \! \left(n +14\right) = -\frac{36 n \left(2 n +3\right) \left(2 n -1\right) a \! \left(n \right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{12 \left(8 n^{3}+55 n^{2}+49 n +5\right) a \! \left(n +1\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{3 \left(751 n^{3}+3352 n^{2}+4843 n +2130\right) a \! \left(n +2\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{3 \left(5257 n^{3}+39890 n^{2}+99971 n +84370\right) a \! \left(n +3\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{3 \left(8381 n^{3}+93920 n^{2}+358973 n +469182\right) a \! \left(n +4\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(32171 n^{3}+491295 n^{2}+2526556 n +4371966\right) a \! \left(n +5\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{\left(173203 n^{3}+3217782 n^{2}+19904909 n +41007258\right) a \! \left(n +6\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(274691 n^{3}+5971644 n^{2}+43126291 n +103470606\right) a \! \left(n +7\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{3 \left(86183 n^{3}+2145884 n^{2}+17750559 n +48777118\right) a \! \left(n +8\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(151685 n^{3}+4252776 n^{2}+39635965 n +122789646\right) a \! \left(n +9\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{\left(56467 n^{3}+1757796 n^{2}+18201671 n +62689674\right) a \! \left(n +10\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{3 \left(2197 n^{3}+75081 n^{2}+853938 n +3232192\right) a \! \left(n +11\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{2 \left(460 n^{3}+17109 n^{2}+211865 n +873462\right) a \! \left(n +12\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(137 n^{2}+3596 n +23571\right) a \! \left(n +13\right)}{2 \left(n +15\right) \left(2 n +29\right)}, \quad n \geq 14\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 70\)
\(\displaystyle a \! \left(6\right) = 253\)
\(\displaystyle a \! \left(7\right) = 936\)
\(\displaystyle a \! \left(8\right) = 3525\)
\(\displaystyle a \! \left(9\right) = 13463\)
\(\displaystyle a \! \left(10\right) = 52008\)
\(\displaystyle a \! \left(11\right) = 202814\)
\(\displaystyle a \! \left(12\right) = 797236\)
\(\displaystyle a \! \left(13\right) = 3155323\)
\(\displaystyle a \! \left(n +14\right) = -\frac{36 n \left(2 n +3\right) \left(2 n -1\right) a \! \left(n \right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{12 \left(8 n^{3}+55 n^{2}+49 n +5\right) a \! \left(n +1\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{3 \left(751 n^{3}+3352 n^{2}+4843 n +2130\right) a \! \left(n +2\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{3 \left(5257 n^{3}+39890 n^{2}+99971 n +84370\right) a \! \left(n +3\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{3 \left(8381 n^{3}+93920 n^{2}+358973 n +469182\right) a \! \left(n +4\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(32171 n^{3}+491295 n^{2}+2526556 n +4371966\right) a \! \left(n +5\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{\left(173203 n^{3}+3217782 n^{2}+19904909 n +41007258\right) a \! \left(n +6\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(274691 n^{3}+5971644 n^{2}+43126291 n +103470606\right) a \! \left(n +7\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{3 \left(86183 n^{3}+2145884 n^{2}+17750559 n +48777118\right) a \! \left(n +8\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(151685 n^{3}+4252776 n^{2}+39635965 n +122789646\right) a \! \left(n +9\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{\left(56467 n^{3}+1757796 n^{2}+18201671 n +62689674\right) a \! \left(n +10\right)}{2 \left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{3 \left(2197 n^{3}+75081 n^{2}+853938 n +3232192\right) a \! \left(n +11\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}-\frac{2 \left(460 n^{3}+17109 n^{2}+211865 n +873462\right) a \! \left(n +12\right)}{\left(n +15\right) \left(n +14\right) \left(2 n +29\right)}+\frac{\left(137 n^{2}+3596 n +23571\right) a \! \left(n +13\right)}{2 \left(n +15\right) \left(2 n +29\right)}, \quad n \geq 14\)
This specification was found using the strategy pack "Point Placements" and has 13 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 13 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
\end{align*}\)