Av(13452, 13524, 13542, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 31452, 31524, 31542)
Counting Sequence
1, 1, 2, 6, 24, 105, 479, 2264, 11014, 54781, 277243, 1423072, 7390868, 38767770, 205082371, ...
Implicit Equation for the Generating Function
\(\displaystyle 3 x^{2} F \left(x
\right)^{4}-x \left(7 x -2\right) F \left(x
\right)^{3}+\left(5 x -4\right) x F \left(x
\right)^{2}+\left(-x^{2}+4 x -1\right) F \! \left(x \right)-x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 105\)
\(\displaystyle a \! \left(6\right) = 479\)
\(\displaystyle a \! \left(7\right) = 2264\)
\(\displaystyle a \! \left(8\right) = 11014\)
\(\displaystyle a \! \left(9\right) = 54781\)
\(\displaystyle a \! \left(10\right) = 277243\)
\(\displaystyle a \! \left(11\right) = 1423072\)
\(\displaystyle a \! \left(12\right) = 7390868\)
\(\displaystyle a \! \left(13\right) = 38767770\)
\(\displaystyle a \! \left(14\right) = 205082371\)
\(\displaystyle a \! \left(15\right) = 1092873903\)
\(\displaystyle a \! \left(16\right) = 5861285512\)
\(\displaystyle a \! \left(17\right) = 31613001003\)
\(\displaystyle a \! \left(18\right) = 171361532780\)
\(\displaystyle a \! \left(19\right) = 933048862378\)
\(\displaystyle a \! \left(n +20\right) = \frac{17 n \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{14 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{51 \left(n +1\right) \left(89 n^{2}+38 n -136\right) a \! \left(n +1\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{3 \left(55013 n^{3}+579531 n^{2}+1725298 n +1566048\right) a \! \left(n +2\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1154927 n^{3}+12489522 n^{2}+44008339 n +50705526\right) a \! \left(n +3\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1350586 n^{3}+18085830 n^{2}+80401646 n +118607025\right) a \! \left(n +4\right)}{112 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(30203209 n^{3}+527621856 n^{2}+3034053839 n +5752317216\right) a \! \left(n +5\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(12385916 n^{3}+254881968 n^{2}+1715994646 n +3793002999\right) a \! \left(n +6\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1730279 n^{3}-46213812 n^{2}-985848521 n -4048292886\right) a \! \left(n +7\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(209904451 n^{3}+5284434081 n^{2}+44335035926 n +123972535656\right) a \! \left(n +8\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(145843697 n^{3}+3837620745 n^{2}+33772023016 n +99506439720\right) a \! \left(n +9\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(40689028 n^{3}+1544738127 n^{2}+18614492399 n +72251381310\right) a \! \left(n +10\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(200384015 n^{3}+7100323410 n^{2}+83100816883 n +321382863588\right) a \! \left(n +11\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(162922858 n^{3}+5908690971 n^{2}+70942611521 n +281756533740\right) a \! \left(n +12\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(36070427 n^{3}+1364469792 n^{2}+17059995535 n +70385229138\right) a \! \left(n +13\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(4195513 n^{3}+168766410 n^{2}+2250519242 n +9940719990\right) a \! \left(n +14\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(2583997 n^{3}+124574688 n^{2}+2021589587 n +11028864648\right) a \! \left(n +15\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(458995 n^{3}+24644715 n^{2}+440393444 n +2618859564\right) a \! \left(n +16\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(548447 n^{3}+29959344 n^{2}+545382121 n +3308471928\right) a \! \left(n +17\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{3 \left(24613 n^{3}+1393108 n^{2}+26282463 n +165274488\right) a \! \left(n +18\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(4637 n^{2}+180800 n +1762323\right) a \! \left(n +19\right)}{112 \left(n +21\right) \left(2 n +41\right)}, \quad n \geq 20\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 105\)
\(\displaystyle a \! \left(6\right) = 479\)
\(\displaystyle a \! \left(7\right) = 2264\)
\(\displaystyle a \! \left(8\right) = 11014\)
\(\displaystyle a \! \left(9\right) = 54781\)
\(\displaystyle a \! \left(10\right) = 277243\)
\(\displaystyle a \! \left(11\right) = 1423072\)
\(\displaystyle a \! \left(12\right) = 7390868\)
\(\displaystyle a \! \left(13\right) = 38767770\)
\(\displaystyle a \! \left(14\right) = 205082371\)
\(\displaystyle a \! \left(15\right) = 1092873903\)
\(\displaystyle a \! \left(16\right) = 5861285512\)
\(\displaystyle a \! \left(17\right) = 31613001003\)
\(\displaystyle a \! \left(18\right) = 171361532780\)
\(\displaystyle a \! \left(19\right) = 933048862378\)
\(\displaystyle a \! \left(n +20\right) = \frac{17 n \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{14 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{51 \left(n +1\right) \left(89 n^{2}+38 n -136\right) a \! \left(n +1\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{3 \left(55013 n^{3}+579531 n^{2}+1725298 n +1566048\right) a \! \left(n +2\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1154927 n^{3}+12489522 n^{2}+44008339 n +50705526\right) a \! \left(n +3\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1350586 n^{3}+18085830 n^{2}+80401646 n +118607025\right) a \! \left(n +4\right)}{112 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(30203209 n^{3}+527621856 n^{2}+3034053839 n +5752317216\right) a \! \left(n +5\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(12385916 n^{3}+254881968 n^{2}+1715994646 n +3793002999\right) a \! \left(n +6\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1730279 n^{3}-46213812 n^{2}-985848521 n -4048292886\right) a \! \left(n +7\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(209904451 n^{3}+5284434081 n^{2}+44335035926 n +123972535656\right) a \! \left(n +8\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(145843697 n^{3}+3837620745 n^{2}+33772023016 n +99506439720\right) a \! \left(n +9\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(40689028 n^{3}+1544738127 n^{2}+18614492399 n +72251381310\right) a \! \left(n +10\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(200384015 n^{3}+7100323410 n^{2}+83100816883 n +321382863588\right) a \! \left(n +11\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(162922858 n^{3}+5908690971 n^{2}+70942611521 n +281756533740\right) a \! \left(n +12\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(36070427 n^{3}+1364469792 n^{2}+17059995535 n +70385229138\right) a \! \left(n +13\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(4195513 n^{3}+168766410 n^{2}+2250519242 n +9940719990\right) a \! \left(n +14\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(2583997 n^{3}+124574688 n^{2}+2021589587 n +11028864648\right) a \! \left(n +15\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(458995 n^{3}+24644715 n^{2}+440393444 n +2618859564\right) a \! \left(n +16\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(548447 n^{3}+29959344 n^{2}+545382121 n +3308471928\right) a \! \left(n +17\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{3 \left(24613 n^{3}+1393108 n^{2}+26282463 n +165274488\right) a \! \left(n +18\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(4637 n^{2}+180800 n +1762323\right) a \! \left(n +19\right)}{112 \left(n +21\right) \left(2 n +41\right)}, \quad n \geq 20\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 31 rules.
Found on January 25, 2022.Finding the specification took 2732 seconds.
Copy 31 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_{11}\! \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_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \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) &= \frac{F_{10}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= \frac{F_{15}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{11}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{0} \left(x \right)^{2} F_{11}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)