Av(13452, 13524, 13542, 15324, 15342, 31452, 31524, 31542)
Counting Sequence
1, 1, 2, 6, 24, 112, 562, 2930, 15666, 85413, 473238, 2657723, 15097742, 86606667, 500992594, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{3} F \left(x
\right)^{6}-x^{2} F \left(x
\right)^{5}-x \left(x -1\right) F \left(x
\right)^{4}-F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 112\)
\(\displaystyle a \! \left(6\right) = 562\)
\(\displaystyle a \! \left(7\right) = 2930\)
\(\displaystyle a \! \left(8\right) = 15666\)
\(\displaystyle a \! \left(9\right) = 85413\)
\(\displaystyle a \! \left(10\right) = 473238\)
\(\displaystyle a \! \left(11\right) = 2657723\)
\(\displaystyle a \! \left(12\right) = 15097742\)
\(\displaystyle a \! \left(13\right) = 86606667\)
\(\displaystyle a \! \left(14\right) = 500992594\)
\(\displaystyle a \! \left(n +15\right) = \frac{109607417194 \left(2 n +5\right) \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{81 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{23 \left(2 n +5\right) \left(2 n +3\right) \left(n +2\right) \left(8076552113 n^{2}+74458617855 n +147946081648\right) a \! \left(n +1\right)}{27 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(2 n +5\right) \left(836084388162 n^{4}+12550699007454 n^{3}+69131429415505 n^{2}+166535703350687 n +148573838304552\right) a \! \left(n +2\right)}{27 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(19248948405247 n^{5}+405669548446356 n^{4}+3416418607482361 n^{3}+14379912932009724 n^{2}+30256142609690752 n +25455546181174560\right) a \! \left(n +3\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(10315127488348 n^{5}+245919311914797 n^{4}+2338109885274574 n^{3}+11071814937980403 n^{2}+26087434604143438 n +24441159653751480\right) a \! \left(n +4\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(2739116556368 n^{5}+61578677026335 n^{4}+494232491341040 n^{3}+1497403412016015 n^{2}+55646912368922 n -5324836641001080\right) a \! \left(n +5\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(167282674871 n^{5}+16856631861441 n^{4}+383383955234429 n^{3}+3677148242087451 n^{2}+16237109802535808 n +27319033397690160\right) a \! \left(n +6\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(907364972123 n^{5}+41850934493976 n^{4}+750621216414665 n^{3}+6596554473451488 n^{2}+28550919404533508 n +48857111387726640\right) a \! \left(n +7\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(141526445979 n^{5}+6416393134437 n^{4}+115961100831607 n^{3}+1044784127344203 n^{2}+4694833407038774 n +8420664858338280\right) a \! \left(n +8\right)}{216 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(120736399883 n^{5}+5690145214302 n^{4}+107308064552669 n^{3}+1012343164818018 n^{2}+4778217975489728 n +9028112752651200\right) a \! \left(n +9\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{5 \left(4410538861 n^{5}+221585688297 n^{4}+4452786138685 n^{3}+44745138217599 n^{2}+224888036767606 n +452350638893064\right) a \! \left(n +10\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(2488546871 n^{5}+135086856996 n^{4}+2931099120029 n^{3}+31780350868356 n^{2}+172209177050228 n +373147890781200\right) a \! \left(n +11\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(57732633 n^{5}+3407543361 n^{4}+80364661945 n^{3}+946731829263 n^{2}+5571307037798 n +13103351073960\right) a \! \left(n +12\right)}{216 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(251803 n^{4}+12433282 n^{3}+230136361 n^{2}+1893027562 n +5840503320\right) a \! \left(n +13\right)}{24 \left(n +15\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(2327 n^{3}+104240 n^{2}+1552741 n +7692748\right) a \! \left(n +14\right)}{24 \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}, \quad n \geq 15\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 112\)
\(\displaystyle a \! \left(6\right) = 562\)
\(\displaystyle a \! \left(7\right) = 2930\)
\(\displaystyle a \! \left(8\right) = 15666\)
\(\displaystyle a \! \left(9\right) = 85413\)
\(\displaystyle a \! \left(10\right) = 473238\)
\(\displaystyle a \! \left(11\right) = 2657723\)
\(\displaystyle a \! \left(12\right) = 15097742\)
\(\displaystyle a \! \left(13\right) = 86606667\)
\(\displaystyle a \! \left(14\right) = 500992594\)
\(\displaystyle a \! \left(n +15\right) = \frac{109607417194 \left(2 n +5\right) \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{81 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{23 \left(2 n +5\right) \left(2 n +3\right) \left(n +2\right) \left(8076552113 n^{2}+74458617855 n +147946081648\right) a \! \left(n +1\right)}{27 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(2 n +5\right) \left(836084388162 n^{4}+12550699007454 n^{3}+69131429415505 n^{2}+166535703350687 n +148573838304552\right) a \! \left(n +2\right)}{27 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(19248948405247 n^{5}+405669548446356 n^{4}+3416418607482361 n^{3}+14379912932009724 n^{2}+30256142609690752 n +25455546181174560\right) a \! \left(n +3\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(10315127488348 n^{5}+245919311914797 n^{4}+2338109885274574 n^{3}+11071814937980403 n^{2}+26087434604143438 n +24441159653751480\right) a \! \left(n +4\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(2739116556368 n^{5}+61578677026335 n^{4}+494232491341040 n^{3}+1497403412016015 n^{2}+55646912368922 n -5324836641001080\right) a \! \left(n +5\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(167282674871 n^{5}+16856631861441 n^{4}+383383955234429 n^{3}+3677148242087451 n^{2}+16237109802535808 n +27319033397690160\right) a \! \left(n +6\right)}{324 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(907364972123 n^{5}+41850934493976 n^{4}+750621216414665 n^{3}+6596554473451488 n^{2}+28550919404533508 n +48857111387726640\right) a \! \left(n +7\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(141526445979 n^{5}+6416393134437 n^{4}+115961100831607 n^{3}+1044784127344203 n^{2}+4694833407038774 n +8420664858338280\right) a \! \left(n +8\right)}{216 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(120736399883 n^{5}+5690145214302 n^{4}+107308064552669 n^{3}+1012343164818018 n^{2}+4778217975489728 n +9028112752651200\right) a \! \left(n +9\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{5 \left(4410538861 n^{5}+221585688297 n^{4}+4452786138685 n^{3}+44745138217599 n^{2}+224888036767606 n +452350638893064\right) a \! \left(n +10\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(2488546871 n^{5}+135086856996 n^{4}+2931099120029 n^{3}+31780350868356 n^{2}+172209177050228 n +373147890781200\right) a \! \left(n +11\right)}{648 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(57732633 n^{5}+3407543361 n^{4}+80364661945 n^{3}+946731829263 n^{2}+5571307037798 n +13103351073960\right) a \! \left(n +12\right)}{216 \left(n +15\right) \left(n +14\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}-\frac{\left(251803 n^{4}+12433282 n^{3}+230136361 n^{2}+1893027562 n +5840503320\right) a \! \left(n +13\right)}{24 \left(n +15\right) \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}+\frac{\left(2327 n^{3}+104240 n^{2}+1552741 n +7692748\right) a \! \left(n +14\right)}{24 \left(3 n +44\right) \left(3 n +46\right) \left(n +16\right)}, \quad n \geq 15\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 47 rules.
Found on January 25, 2022.Finding the specification took 1226 seconds.
Copy 47 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_{19}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{11}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{11}\! \left(x \right) F_{41}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{11}\! \left(x \right) F_{36}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{0}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{11}\! \left(x \right) F_{39}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= \frac{F_{40}\! \left(x \right)}{F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right)\\
\end{align*}\)