Av(12453, 12543, 14253, 21453, 21543, 24153, 24513, 25143, 25413, 41253, 42153, 42513)
Counting Sequence
1, 1, 2, 6, 24, 108, 512, 2501, 12469, 63117, 323302, 1671958, 8715360, 45736091, 241402304, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -2\right)^{3} F \left(x
\right)^{4}-\left(x -2\right) \left(2 x^{3}-5 x^{2}+7 x +2\right) F \left(x
\right)^{3}+\left(x^{4}-4 x^{3}-x^{2}+5 x -12\right) F \left(x
\right)^{2}+\left(x^{3}+6 x^{2}-13 x +12\right) F \! \left(x \right)-\left(x -2\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 108\)
\(\displaystyle a \! \left(6\right) = 512\)
\(\displaystyle a \! \left(7\right) = 2501\)
\(\displaystyle a \! \left(8\right) = 12469\)
\(\displaystyle a \! \left(9\right) = 63117\)
\(\displaystyle a \! \left(10\right) = 323302\)
\(\displaystyle a \! \left(11\right) = 1671958\)
\(\displaystyle a \! \left(12\right) = 8715360\)
\(\displaystyle a \! \left(13\right) = 45736091\)
\(\displaystyle a \! \left(14\right) = 241402304\)
\(\displaystyle a \! \left(15\right) = 1280603408\)
\(\displaystyle a \! \left(16\right) = 6823766712\)
\(\displaystyle a \! \left(17\right) = 36505829481\)
\(\displaystyle a \! \left(18\right) = 196000221628\)
\(\displaystyle a \! \left(19\right) = 1055753103958\)
\(\displaystyle a \! \left(20\right) = 5703695476643\)
\(\displaystyle a \! \left(21\right) = 30898137957678\)
\(\displaystyle a \! \left(22\right) = 167802344738747\)
\(\displaystyle a \! \left(23\right) = 913424294068838\)
\(\displaystyle a \! \left(24\right) = 4982925370558436\)
\(\displaystyle a \! \left(25\right) = 27237662753631018\)
\(\displaystyle a \! \left(26\right) = 149166709507890233\)
\(\displaystyle a \! \left(27\right) = 818350303123129654\)
\(\displaystyle a \! \left(n +28\right) = -\frac{\left(531001369961 n^{3}+26306786644710 n^{2}+433489777776679 n +2375586259876506\right) a \! \left(n +17\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(307298183668 n^{3}+15912614787105 n^{2}+273706497937685 n +1563295725282534\right) a \! \left(n +18\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(741983406041 n^{3}+32890936371861 n^{2}+485448799376962 n +2385545264638644\right) a \! \left(n +15\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(700784092055 n^{3}+32941685418648 n^{2}+515374537405225 n +2683433893055406\right) a \! \left(n +16\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(328599474 n^{3}+5972655641 n^{2}+36279549427 n +73639719230\right) a \! \left(n +6\right)}{10240 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(8502071 n^{3}+638684184 n^{2}+15988566751 n +133383489654\right) a \! \left(n +25\right)}{960 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(969455 n^{3}+75521514 n^{2}+1960413373 n +16957668738\right) a \! \left(n +26\right)}{960 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(8069 n^{3}+651303 n^{2}+17516788 n +156978780\right) a \! \left(n +27\right)}{120 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(237664284106 n^{3}+9197509610028 n^{2}+118578908298797 n +509313370025646\right) a \! \left(n +13\right)}{15360 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(647849616911 n^{3}+26920950651375 n^{2}+372579311384284 n +1717360306380432\right) a \! \left(n +14\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(120507824759 n^{3}+6398751887964 n^{2}+112355150114875 n +651437342346234\right) a \! \left(n +19\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(8552975023 n^{3}+388012621785 n^{2}+5187819208514 n +16996336585704\right) a \! \left(n +20\right)}{15360 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(14877233195 n^{3}+1037532442962 n^{2}+23941077793165 n +182972002172190\right) a \! \left(n +21\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(31516778278 n^{3}+841462557981 n^{2}+7496579138537 n +22287597464106\right) a \! \left(n +9\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(297483730202 n^{3}+10640720094825 n^{2}+126836603311975 n +503847945646062\right) a \! \left(n +12\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(75974323358 n^{3}+2258935692363 n^{2}+22401001460167 n +74094508634898\right) a \! \left(n +10\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(160879333307 n^{3}+5272307058660 n^{2}+57601354945165 n +209805861629610\right) a \! \left(n +11\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(2189485589 n^{3}+149351061358 n^{2}+3393952568051 n +25693998013522\right) a \! \left(n +22\right)}{5120 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(468972551 n^{3}+32838058058 n^{2}+766326045129 n +5960132903878\right) a \! \left(n +23\right)}{2560 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(47712497 n^{3}+3456891525 n^{2}+83470483981 n +671705393211\right) a \! \left(n +24\right)}{960 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{5 \left(n +2\right) \left(12219 n^{2}+78434 n +124405\right) a \! \left(n +2\right)}{2048 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(1210250547 n^{3}+25267196910 n^{2}+176205532001 n +410472808158\right) a \! \left(n +7\right)}{10240 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(955935121 n^{3}+22683724790 n^{2}+179700955093 n +475308321231\right) a \! \left(n +8\right)}{2560 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{5 \left(283018 n^{3}+3076239 n^{2}+11053949 n +13119282\right) a \! \left(n +3\right)}{6144 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(4386446 n^{3}+58007511 n^{2}+255672610 n +375332838\right) a \! \left(n +4\right)}{3072 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{875 n \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{6144 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{25 \left(228 n +625\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{2048 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(22674653 n^{3}+354467859 n^{2}+1851588214 n +3230615070\right) a \! \left(n +5\right)}{3072 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}, \quad n \geq 28\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 108\)
\(\displaystyle a \! \left(6\right) = 512\)
\(\displaystyle a \! \left(7\right) = 2501\)
\(\displaystyle a \! \left(8\right) = 12469\)
\(\displaystyle a \! \left(9\right) = 63117\)
\(\displaystyle a \! \left(10\right) = 323302\)
\(\displaystyle a \! \left(11\right) = 1671958\)
\(\displaystyle a \! \left(12\right) = 8715360\)
\(\displaystyle a \! \left(13\right) = 45736091\)
\(\displaystyle a \! \left(14\right) = 241402304\)
\(\displaystyle a \! \left(15\right) = 1280603408\)
\(\displaystyle a \! \left(16\right) = 6823766712\)
\(\displaystyle a \! \left(17\right) = 36505829481\)
\(\displaystyle a \! \left(18\right) = 196000221628\)
\(\displaystyle a \! \left(19\right) = 1055753103958\)
\(\displaystyle a \! \left(20\right) = 5703695476643\)
\(\displaystyle a \! \left(21\right) = 30898137957678\)
\(\displaystyle a \! \left(22\right) = 167802344738747\)
\(\displaystyle a \! \left(23\right) = 913424294068838\)
\(\displaystyle a \! \left(24\right) = 4982925370558436\)
\(\displaystyle a \! \left(25\right) = 27237662753631018\)
\(\displaystyle a \! \left(26\right) = 149166709507890233\)
\(\displaystyle a \! \left(27\right) = 818350303123129654\)
\(\displaystyle a \! \left(n +28\right) = -\frac{\left(531001369961 n^{3}+26306786644710 n^{2}+433489777776679 n +2375586259876506\right) a \! \left(n +17\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(307298183668 n^{3}+15912614787105 n^{2}+273706497937685 n +1563295725282534\right) a \! \left(n +18\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(741983406041 n^{3}+32890936371861 n^{2}+485448799376962 n +2385545264638644\right) a \! \left(n +15\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(700784092055 n^{3}+32941685418648 n^{2}+515374537405225 n +2683433893055406\right) a \! \left(n +16\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(328599474 n^{3}+5972655641 n^{2}+36279549427 n +73639719230\right) a \! \left(n +6\right)}{10240 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(8502071 n^{3}+638684184 n^{2}+15988566751 n +133383489654\right) a \! \left(n +25\right)}{960 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(969455 n^{3}+75521514 n^{2}+1960413373 n +16957668738\right) a \! \left(n +26\right)}{960 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(8069 n^{3}+651303 n^{2}+17516788 n +156978780\right) a \! \left(n +27\right)}{120 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(237664284106 n^{3}+9197509610028 n^{2}+118578908298797 n +509313370025646\right) a \! \left(n +13\right)}{15360 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(647849616911 n^{3}+26920950651375 n^{2}+372579311384284 n +1717360306380432\right) a \! \left(n +14\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(120507824759 n^{3}+6398751887964 n^{2}+112355150114875 n +651437342346234\right) a \! \left(n +19\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(8552975023 n^{3}+388012621785 n^{2}+5187819208514 n +16996336585704\right) a \! \left(n +20\right)}{15360 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(14877233195 n^{3}+1037532442962 n^{2}+23941077793165 n +182972002172190\right) a \! \left(n +21\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(31516778278 n^{3}+841462557981 n^{2}+7496579138537 n +22287597464106\right) a \! \left(n +9\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(297483730202 n^{3}+10640720094825 n^{2}+126836603311975 n +503847945646062\right) a \! \left(n +12\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(75974323358 n^{3}+2258935692363 n^{2}+22401001460167 n +74094508634898\right) a \! \left(n +10\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(160879333307 n^{3}+5272307058660 n^{2}+57601354945165 n +209805861629610\right) a \! \left(n +11\right)}{30720 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(2189485589 n^{3}+149351061358 n^{2}+3393952568051 n +25693998013522\right) a \! \left(n +22\right)}{5120 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(468972551 n^{3}+32838058058 n^{2}+766326045129 n +5960132903878\right) a \! \left(n +23\right)}{2560 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(47712497 n^{3}+3456891525 n^{2}+83470483981 n +671705393211\right) a \! \left(n +24\right)}{960 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{5 \left(n +2\right) \left(12219 n^{2}+78434 n +124405\right) a \! \left(n +2\right)}{2048 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(1210250547 n^{3}+25267196910 n^{2}+176205532001 n +410472808158\right) a \! \left(n +7\right)}{10240 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(955935121 n^{3}+22683724790 n^{2}+179700955093 n +475308321231\right) a \! \left(n +8\right)}{2560 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{5 \left(283018 n^{3}+3076239 n^{2}+11053949 n +13119282\right) a \! \left(n +3\right)}{6144 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{\left(4386446 n^{3}+58007511 n^{2}+255672610 n +375332838\right) a \! \left(n +4\right)}{3072 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}+\frac{875 n \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{6144 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{25 \left(228 n +625\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{2048 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}-\frac{\left(22674653 n^{3}+354467859 n^{2}+1851588214 n +3230615070\right) a \! \left(n +5\right)}{3072 \left(2 n +55\right) \left(n +29\right) \left(n +27\right)}, \quad n \geq 28\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 70 rules.
Found on January 23, 2022.Finding the specification took 145 seconds.
Copy 70 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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{36}\! \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) &= \frac{F_{11}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\
F_{24}\! \left(x , y\right) &= \frac{F_{25}\! \left(x , y\right) y -F_{25}\! \left(x , 1\right)}{-1+y}\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{29}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= y x\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{35}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= \frac{F_{25}\! \left(x , y\right) y -F_{25}\! \left(x , 1\right)}{-1+y}\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{0}\! \left(x \right) F_{38}\! \left(x \right) F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{0}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= \frac{F_{46}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x , 1\right)\\
F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right) F_{67}\! \left(x , y\right)\\
F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{18}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= \frac{F_{60}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{67}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{69}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
\end{align*}\)