Av(13452, 13524, 13542, 14352, 14532, 15324, 15342, 15432, 31452, 31524, 31542)
Counting Sequence
1, 1, 2, 6, 24, 109, 522, 2584, 13140, 68320, 361656, 1942474, 10558944, 57978138, 321105516, ...
Implicit Equation for the Generating Function
\(\displaystyle 2 x^{2} F \left(x
\right)^{6}-4 x^{2} F \left(x
\right)^{5}+x \left(2 x +1\right) F \left(x
\right)^{4}-2 x F \left(x
\right)^{3}+2 x F \left(x
\right)^{2}-F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 109\)
\(\displaystyle a \! \left(6\right) = 522\)
\(\displaystyle a \! \left(7\right) = 2584\)
\(\displaystyle a \! \left(8\right) = 13140\)
\(\displaystyle a \! \left(9\right) = 68320\)
\(\displaystyle a \! \left(10\right) = 361656\)
\(\displaystyle a \! \left(11\right) = 1942474\)
\(\displaystyle a \! \left(12\right) = 10558944\)
\(\displaystyle a \! \left(13\right) = 57978138\)
\(\displaystyle a \! \left(14\right) = 321105516\)
\(\displaystyle a \! \left(15\right) = 1791691792\)
\(\displaystyle a \! \left(16\right) = 10062353548\)
\(\displaystyle a \! \left(17\right) = 56835384632\)
\(\displaystyle a \! \left(18\right) = 322656494440\)
\(\displaystyle a \! \left(19\right) = 1840046669608\)
\(\displaystyle a \! \left(n +20\right) = -\frac{66538438656 n \left(2 n +5\right) \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) a \! \left(n \right)}{115 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{25165824 \left(2 n +5\right) \left(2 n +3\right) \left(826051 n^{3}+5251606 n^{2}+10766816 n +6331346\right) a \! \left(n +1\right)}{575 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{1048576 \left(2 n +5\right) \left(419744543 n^{4}+4753917485 n^{3}+20140200060 n^{2}+37657506080 n +26055162252\right) a \! \left(n +2\right)}{575 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{524288 \left(56243430352 n^{5}+950757418235 n^{4}+6347953992500 n^{3}+20877992358460 n^{2}+33734532761073 n +21351583420020\right) a \! \left(n +3\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{131072 \left(4624106518 n^{5}+204906948227 n^{4}+3077030850953 n^{3}+20468514358942 n^{2}+62591864905626 n +72072677034504\right) a \! \left(n +4\right)}{1035 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{65536 \left(459633013454 n^{5}+10686494860300 n^{4}+95044199592190 n^{3}+394611961602135 n^{2}+723639454901441 n +390570423733230\right) a \! \left(n +5\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{16384 \left(22252428179902 n^{5}+728772702954545 n^{4}+9602196417200105 n^{3}+63548559300612190 n^{2}+211012237923911778 n +280942969435624620\right) a \! \left(n +6\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{8192 \left(463954738906 n^{5}+58496329410920 n^{4}+1442525360983670 n^{3}+14704404926064115 n^{2}+68488421770314069 n +121032704218180260\right) a \! \left(n +7\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{2048 \left(75296589067046 n^{5}+3353733823804525 n^{4}+59012671074038755 n^{3}+513914288005106330 n^{2}+2218465475940804294 n +3802085173860464160\right) a \! \left(n +8\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{1024 \left(175835199949456 n^{5}+8210004220322225 n^{4}+153573772992273440 n^{3}+1438564975825879030 n^{2}+6747661796330515539 n +12677442553213349670\right) a \! \left(n +9\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{512 \left(47691619747724 n^{5}+2391125493634915 n^{4}+47926522426461715 n^{3}+479845148252368895 n^{2}+2398808567643373101 n +4788003291362148360\right) a \! \left(n +10\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{512 \left(2780164758233 n^{5}+150213574335214 n^{4}+3237147465706754 n^{3}+34777507467601862 n^{2}+186244210729413050 n +397724882570176239\right) a \! \left(n +11\right)}{345 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{256 \left(9541083974626 n^{5}+557002188356735 n^{4}+13033073349114380 n^{3}+152895503433900160 n^{2}+900011606739133074 n +2128467569837601510\right) a \! \left(n +12\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{64 \left(547852803396 n^{5}+37403673002705 n^{4}+1018627112613550 n^{3}+13833629275830135 n^{2}+93695354481967224 n +253206952707696630\right) a \! \left(n +13\right)}{575 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{64 \left(322835506114 n^{5}+22116001056755 n^{4}+604189699634700 n^{3}+8223943792908330 n^{2}+55740844638683286 n +150395619122828460\right) a \! \left(n +14\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{16 \left(257044162364 n^{5}+21601032477325 n^{4}+720168677105740 n^{3}+11920703915177375 n^{2}+98055655743052206 n +320886118254233790\right) a \! \left(n +15\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{8 \left(49219463984 n^{5}+3961387765405 n^{4}+127471130728165 n^{3}+2049913351279910 n^{2}+16474550727907356 n +52933591129761180\right) a \! \left(n +16\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{4 \left(4137428896 n^{5}+372951527030 n^{4}+13419854507630 n^{3}+240995080593505 n^{2}+2160254180206539 n +7733761194695400\right) a \! \left(n +17\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{2 \left(423777506 n^{5}+38903450095 n^{4}+1427528408905 n^{3}+26171679899375 n^{2}+239729565013239 n +877681300453200\right) a \! \left(n +18\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{\left(1423042 n^{4}+110371909 n^{3}+3211837189 n^{2}+41561377584 n +201781947072\right) a \! \left(n +19\right)}{345 \left(n +20\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}, \quad n \geq 20\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 109\)
\(\displaystyle a \! \left(6\right) = 522\)
\(\displaystyle a \! \left(7\right) = 2584\)
\(\displaystyle a \! \left(8\right) = 13140\)
\(\displaystyle a \! \left(9\right) = 68320\)
\(\displaystyle a \! \left(10\right) = 361656\)
\(\displaystyle a \! \left(11\right) = 1942474\)
\(\displaystyle a \! \left(12\right) = 10558944\)
\(\displaystyle a \! \left(13\right) = 57978138\)
\(\displaystyle a \! \left(14\right) = 321105516\)
\(\displaystyle a \! \left(15\right) = 1791691792\)
\(\displaystyle a \! \left(16\right) = 10062353548\)
\(\displaystyle a \! \left(17\right) = 56835384632\)
\(\displaystyle a \! \left(18\right) = 322656494440\)
\(\displaystyle a \! \left(19\right) = 1840046669608\)
\(\displaystyle a \! \left(n +20\right) = -\frac{66538438656 n \left(2 n +5\right) \left(2 n +3\right) \left(2 n +1\right) \left(n +2\right) a \! \left(n \right)}{115 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{25165824 \left(2 n +5\right) \left(2 n +3\right) \left(826051 n^{3}+5251606 n^{2}+10766816 n +6331346\right) a \! \left(n +1\right)}{575 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{1048576 \left(2 n +5\right) \left(419744543 n^{4}+4753917485 n^{3}+20140200060 n^{2}+37657506080 n +26055162252\right) a \! \left(n +2\right)}{575 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{524288 \left(56243430352 n^{5}+950757418235 n^{4}+6347953992500 n^{3}+20877992358460 n^{2}+33734532761073 n +21351583420020\right) a \! \left(n +3\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{131072 \left(4624106518 n^{5}+204906948227 n^{4}+3077030850953 n^{3}+20468514358942 n^{2}+62591864905626 n +72072677034504\right) a \! \left(n +4\right)}{1035 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{65536 \left(459633013454 n^{5}+10686494860300 n^{4}+95044199592190 n^{3}+394611961602135 n^{2}+723639454901441 n +390570423733230\right) a \! \left(n +5\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{16384 \left(22252428179902 n^{5}+728772702954545 n^{4}+9602196417200105 n^{3}+63548559300612190 n^{2}+211012237923911778 n +280942969435624620\right) a \! \left(n +6\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{8192 \left(463954738906 n^{5}+58496329410920 n^{4}+1442525360983670 n^{3}+14704404926064115 n^{2}+68488421770314069 n +121032704218180260\right) a \! \left(n +7\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{2048 \left(75296589067046 n^{5}+3353733823804525 n^{4}+59012671074038755 n^{3}+513914288005106330 n^{2}+2218465475940804294 n +3802085173860464160\right) a \! \left(n +8\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{1024 \left(175835199949456 n^{5}+8210004220322225 n^{4}+153573772992273440 n^{3}+1438564975825879030 n^{2}+6747661796330515539 n +12677442553213349670\right) a \! \left(n +9\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{512 \left(47691619747724 n^{5}+2391125493634915 n^{4}+47926522426461715 n^{3}+479845148252368895 n^{2}+2398808567643373101 n +4788003291362148360\right) a \! \left(n +10\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{512 \left(2780164758233 n^{5}+150213574335214 n^{4}+3237147465706754 n^{3}+34777507467601862 n^{2}+186244210729413050 n +397724882570176239\right) a \! \left(n +11\right)}{345 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{256 \left(9541083974626 n^{5}+557002188356735 n^{4}+13033073349114380 n^{3}+152895503433900160 n^{2}+900011606739133074 n +2128467569837601510\right) a \! \left(n +12\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{64 \left(547852803396 n^{5}+37403673002705 n^{4}+1018627112613550 n^{3}+13833629275830135 n^{2}+93695354481967224 n +253206952707696630\right) a \! \left(n +13\right)}{575 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{64 \left(322835506114 n^{5}+22116001056755 n^{4}+604189699634700 n^{3}+8223943792908330 n^{2}+55740844638683286 n +150395619122828460\right) a \! \left(n +14\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{16 \left(257044162364 n^{5}+21601032477325 n^{4}+720168677105740 n^{3}+11920703915177375 n^{2}+98055655743052206 n +320886118254233790\right) a \! \left(n +15\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{8 \left(49219463984 n^{5}+3961387765405 n^{4}+127471130728165 n^{3}+2049913351279910 n^{2}+16474550727907356 n +52933591129761180\right) a \! \left(n +16\right)}{1725 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}+\frac{4 \left(4137428896 n^{5}+372951527030 n^{4}+13419854507630 n^{3}+240995080593505 n^{2}+2160254180206539 n +7733761194695400\right) a \! \left(n +17\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{2 \left(423777506 n^{5}+38903450095 n^{4}+1427528408905 n^{3}+26171679899375 n^{2}+239729565013239 n +877681300453200\right) a \! \left(n +18\right)}{5175 \left(n +20\right) \left(n +19\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}-\frac{\left(1423042 n^{4}+110371909 n^{3}+3211837189 n^{2}+41561377584 n +201781947072\right) a \! \left(n +19\right)}{345 \left(n +20\right) \left(3 n +59\right) \left(2 n +41\right) \left(3 n +61\right)}, \quad n \geq 20\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 66 rules.
Found on January 25, 2022.Finding the specification took 2058 seconds.
Copy 66 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_{21}\! \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_{21}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{27}\! \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) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{21}\! \left(x \right) &= x\\
F_{22}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{0}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{21}\! \left(x \right) F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{21}\! \left(x \right) F_{32}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{21}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{21}\! \left(x \right) F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{21}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{21}\! \left(x \right) F_{51}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{21}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{0}\! \left(x \right) F_{21}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{52}\! \left(x \right) &= \frac{F_{53}\! \left(x \right)}{F_{0}\! \left(x \right) F_{21}\! \left(x \right) F_{38}\! \left(x \right)}\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{0}\! \left(x \right) F_{21}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{0}\! \left(x \right) F_{21}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{38} \left(x \right)^{2} F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= -F_{9}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{21}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{63}\! \left(x \right)\\
\end{align*}\)