Av(12354, 12453, 13254, 21354, 21453, 23154, 31254, 31452, 32154, 41253, 41352, 42153)
Counting Sequence
1, 1, 2, 6, 24, 108, 512, 2501, 12469, 63117, 323302, 1671958, 8715360, 45736091, 241402304, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(-2+x \right)^{3} F \left(x
\right)^{4}-\left(-2+x \right) \left(2 x^{3}-5 x^{2}+7 x +2\right) F \left(x
\right)^{3}+\left(x^{4}-4 x^{3}-x^{2}+5 x -12\right) F \left(x
\right)^{2}+\left(x^{3}+6 x^{2}-13 x +12\right) F \! \left(x \right)-\left(-2+x \right)^{2} = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 108\)
\(\displaystyle a(6) = 512\)
\(\displaystyle a(7) = 2501\)
\(\displaystyle a(8) = 12469\)
\(\displaystyle a(9) = 63117\)
\(\displaystyle a(10) = 323302\)
\(\displaystyle a(11) = 1671958\)
\(\displaystyle a(12) = 8715360\)
\(\displaystyle a(13) = 45736091\)
\(\displaystyle a(14) = 241402304\)
\(\displaystyle a(15) = 1280603408\)
\(\displaystyle a(16) = 6823766712\)
\(\displaystyle a(17) = 36505829481\)
\(\displaystyle a(18) = 196000221628\)
\(\displaystyle a(19) = 1055753103958\)
\(\displaystyle a(20) = 5703695476643\)
\(\displaystyle a(21) = 30898137957678\)
\(\displaystyle a(22) = 167802344738747\)
\(\displaystyle a(23) = 913424294068838\)
\(\displaystyle a(24) = 4982925370558436\)
\(\displaystyle a(25) = 27237662753631018\)
\(\displaystyle a(26) = 149166709507890233\)
\(\displaystyle a(27) = 818350303123129654\)
\(\displaystyle a{\left(n + 28 \right)} = \frac{875 n \left(n + 1\right) \left(n + 2\right) a{\left(n \right)}}{6144 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{25 \left(n + 1\right) \left(n + 2\right) \left(228 n + 625\right) a{\left(n + 1 \right)}}{2048 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{5 \left(n + 2\right) \left(12219 n^{2} + 78434 n + 124405\right) a{\left(n + 2 \right)}}{2048 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(8069 n^{3} + 651303 n^{2} + 17516788 n + 156978780\right) a{\left(n + 27 \right)}}{120 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{5 \left(283018 n^{3} + 3076239 n^{2} + 11053949 n + 13119282\right) a{\left(n + 3 \right)}}{6144 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(969455 n^{3} + 75521514 n^{2} + 1960413373 n + 16957668738\right) a{\left(n + 26 \right)}}{960 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(4386446 n^{3} + 58007511 n^{2} + 255672610 n + 375332838\right) a{\left(n + 4 \right)}}{3072 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(8502071 n^{3} + 638684184 n^{2} + 15988566751 n + 133383489654\right) a{\left(n + 25 \right)}}{960 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(22674653 n^{3} + 354467859 n^{2} + 1851588214 n + 3230615070\right) a{\left(n + 5 \right)}}{3072 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(47712497 n^{3} + 3456891525 n^{2} + 83470483981 n + 671705393211\right) a{\left(n + 24 \right)}}{960 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(328599474 n^{3} + 5972655641 n^{2} + 36279549427 n + 73639719230\right) a{\left(n + 6 \right)}}{10240 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(468972551 n^{3} + 32838058058 n^{2} + 766326045129 n + 5960132903878\right) a{\left(n + 23 \right)}}{2560 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(955935121 n^{3} + 22683724790 n^{2} + 179700955093 n + 475308321231\right) a{\left(n + 8 \right)}}{2560 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(1210250547 n^{3} + 25267196910 n^{2} + 176205532001 n + 410472808158\right) a{\left(n + 7 \right)}}{10240 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(2189485589 n^{3} + 149351061358 n^{2} + 3393952568051 n + 25693998013522\right) a{\left(n + 22 \right)}}{5120 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(8552975023 n^{3} + 388012621785 n^{2} + 5187819208514 n + 16996336585704\right) a{\left(n + 20 \right)}}{15360 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(14877233195 n^{3} + 1037532442962 n^{2} + 23941077793165 n + 182972002172190\right) a{\left(n + 21 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(31516778278 n^{3} + 841462557981 n^{2} + 7496579138537 n + 22287597464106\right) a{\left(n + 9 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(75974323358 n^{3} + 2258935692363 n^{2} + 22401001460167 n + 74094508634898\right) a{\left(n + 10 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(120507824759 n^{3} + 6398751887964 n^{2} + 112355150114875 n + 651437342346234\right) a{\left(n + 19 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(160879333307 n^{3} + 5272307058660 n^{2} + 57601354945165 n + 209805861629610\right) a{\left(n + 11 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(237664284106 n^{3} + 9197509610028 n^{2} + 118578908298797 n + 509313370025646\right) a{\left(n + 13 \right)}}{15360 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(297483730202 n^{3} + 10640720094825 n^{2} + 126836603311975 n + 503847945646062\right) a{\left(n + 12 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(307298183668 n^{3} + 15912614787105 n^{2} + 273706497937685 n + 1563295725282534\right) a{\left(n + 18 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(531001369961 n^{3} + 26306786644710 n^{2} + 433489777776679 n + 2375586259876506\right) a{\left(n + 17 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(647849616911 n^{3} + 26920950651375 n^{2} + 372579311384284 n + 1717360306380432\right) a{\left(n + 14 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(700784092055 n^{3} + 32941685418648 n^{2} + 515374537405225 n + 2683433893055406\right) a{\left(n + 16 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(741983406041 n^{3} + 32890936371861 n^{2} + 485448799376962 n + 2385545264638644\right) a{\left(n + 15 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)}, \quad n \geq 28\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 108\)
\(\displaystyle a(6) = 512\)
\(\displaystyle a(7) = 2501\)
\(\displaystyle a(8) = 12469\)
\(\displaystyle a(9) = 63117\)
\(\displaystyle a(10) = 323302\)
\(\displaystyle a(11) = 1671958\)
\(\displaystyle a(12) = 8715360\)
\(\displaystyle a(13) = 45736091\)
\(\displaystyle a(14) = 241402304\)
\(\displaystyle a(15) = 1280603408\)
\(\displaystyle a(16) = 6823766712\)
\(\displaystyle a(17) = 36505829481\)
\(\displaystyle a(18) = 196000221628\)
\(\displaystyle a(19) = 1055753103958\)
\(\displaystyle a(20) = 5703695476643\)
\(\displaystyle a(21) = 30898137957678\)
\(\displaystyle a(22) = 167802344738747\)
\(\displaystyle a(23) = 913424294068838\)
\(\displaystyle a(24) = 4982925370558436\)
\(\displaystyle a(25) = 27237662753631018\)
\(\displaystyle a(26) = 149166709507890233\)
\(\displaystyle a(27) = 818350303123129654\)
\(\displaystyle a{\left(n + 28 \right)} = \frac{875 n \left(n + 1\right) \left(n + 2\right) a{\left(n \right)}}{6144 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{25 \left(n + 1\right) \left(n + 2\right) \left(228 n + 625\right) a{\left(n + 1 \right)}}{2048 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{5 \left(n + 2\right) \left(12219 n^{2} + 78434 n + 124405\right) a{\left(n + 2 \right)}}{2048 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(8069 n^{3} + 651303 n^{2} + 17516788 n + 156978780\right) a{\left(n + 27 \right)}}{120 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{5 \left(283018 n^{3} + 3076239 n^{2} + 11053949 n + 13119282\right) a{\left(n + 3 \right)}}{6144 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(969455 n^{3} + 75521514 n^{2} + 1960413373 n + 16957668738\right) a{\left(n + 26 \right)}}{960 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(4386446 n^{3} + 58007511 n^{2} + 255672610 n + 375332838\right) a{\left(n + 4 \right)}}{3072 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(8502071 n^{3} + 638684184 n^{2} + 15988566751 n + 133383489654\right) a{\left(n + 25 \right)}}{960 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(22674653 n^{3} + 354467859 n^{2} + 1851588214 n + 3230615070\right) a{\left(n + 5 \right)}}{3072 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(47712497 n^{3} + 3456891525 n^{2} + 83470483981 n + 671705393211\right) a{\left(n + 24 \right)}}{960 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(328599474 n^{3} + 5972655641 n^{2} + 36279549427 n + 73639719230\right) a{\left(n + 6 \right)}}{10240 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(468972551 n^{3} + 32838058058 n^{2} + 766326045129 n + 5960132903878\right) a{\left(n + 23 \right)}}{2560 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(955935121 n^{3} + 22683724790 n^{2} + 179700955093 n + 475308321231\right) a{\left(n + 8 \right)}}{2560 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(1210250547 n^{3} + 25267196910 n^{2} + 176205532001 n + 410472808158\right) a{\left(n + 7 \right)}}{10240 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(2189485589 n^{3} + 149351061358 n^{2} + 3393952568051 n + 25693998013522\right) a{\left(n + 22 \right)}}{5120 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(8552975023 n^{3} + 388012621785 n^{2} + 5187819208514 n + 16996336585704\right) a{\left(n + 20 \right)}}{15360 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(14877233195 n^{3} + 1037532442962 n^{2} + 23941077793165 n + 182972002172190\right) a{\left(n + 21 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(31516778278 n^{3} + 841462557981 n^{2} + 7496579138537 n + 22287597464106\right) a{\left(n + 9 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(75974323358 n^{3} + 2258935692363 n^{2} + 22401001460167 n + 74094508634898\right) a{\left(n + 10 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(120507824759 n^{3} + 6398751887964 n^{2} + 112355150114875 n + 651437342346234\right) a{\left(n + 19 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(160879333307 n^{3} + 5272307058660 n^{2} + 57601354945165 n + 209805861629610\right) a{\left(n + 11 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(237664284106 n^{3} + 9197509610028 n^{2} + 118578908298797 n + 509313370025646\right) a{\left(n + 13 \right)}}{15360 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(297483730202 n^{3} + 10640720094825 n^{2} + 126836603311975 n + 503847945646062\right) a{\left(n + 12 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(307298183668 n^{3} + 15912614787105 n^{2} + 273706497937685 n + 1563295725282534\right) a{\left(n + 18 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(531001369961 n^{3} + 26306786644710 n^{2} + 433489777776679 n + 2375586259876506\right) a{\left(n + 17 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(647849616911 n^{3} + 26920950651375 n^{2} + 372579311384284 n + 1717360306380432\right) a{\left(n + 14 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} + \frac{\left(700784092055 n^{3} + 32941685418648 n^{2} + 515374537405225 n + 2683433893055406\right) a{\left(n + 16 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)} - \frac{\left(741983406041 n^{3} + 32890936371861 n^{2} + 485448799376962 n + 2385545264638644\right) a{\left(n + 15 \right)}}{30720 \left(n + 27\right) \left(n + 29\right) \left(2 n + 55\right)}, \quad n \geq 28\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 86 rules.
Finding the specification took 2954 seconds.
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\(\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_{15}\! \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_{15}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= \frac{F_{8}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= -F_{84}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{36}\! \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_{15}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{32}\! \left(x \right) &= x^{2} F_{32} \left(x \right)^{2}+2 x^{2} F_{32}\! \left(x \right)-2 x F_{32} \left(x \right)^{2}+x^{2}-3 x F_{32}\! \left(x \right)-x +2 F_{32}\! \left(x \right)\\
F_{33}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{15}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{15}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{44}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{15}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right) F_{50}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{15}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right) F_{58}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= \frac{F_{64}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= \frac{F_{71}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{71}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{15}\! \left(x \right) F_{44}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= -F_{81}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= \frac{F_{80}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{80}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= \frac{F_{83}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{83}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{38}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 82 rules.
Finding the specification took 1579 seconds.
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\(\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_{15}\! \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_{15}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= \frac{F_{8}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= -F_{80}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \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_{15}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{32}\! \left(x \right) &= x^{2} F_{32} \left(x \right)^{2}-2 x F_{32} \left(x \right)^{2}+x F_{32}\! \left(x \right)+2 F_{32}\! \left(x \right)-1\\
F_{33}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= x^{2} F_{37} \left(x \right)^{2}+2 x^{2} F_{37}\! \left(x \right)-2 x F_{37} \left(x \right)^{2}+x^{2}-3 x F_{37}\! \left(x \right)-x +2 F_{37}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{15}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= \frac{F_{71}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{71}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{62}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{79}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{40}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 88 rules.
Finding the specification took 1433 seconds.
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Copy 88 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_{15}\! \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_{15}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= \frac{F_{8}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= -F_{86}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \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_{15}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{32}\! \left(x \right) &= x^{2} F_{32} \left(x \right)^{2}-2 x F_{32} \left(x \right)^{2}+F_{32}\! \left(x \right) x +2 F_{32}\! \left(x \right)-1\\
F_{33}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= x^{2} F_{37} \left(x \right)^{2}+2 x^{2} F_{37}\! \left(x \right)-2 x F_{37} \left(x \right)^{2}+x^{2}-3 x F_{37}\! \left(x \right)-x +2 F_{37}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{15}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{68}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{15}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{79}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{40}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 88 rules.
Finding the specification took 1433 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 88 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_{15}\! \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_{15}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= \frac{F_{8}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= -F_{86}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \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_{15}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{32}\! \left(x \right) &= x^{2} F_{32} \left(x \right)^{2}-2 x F_{32} \left(x \right)^{2}+F_{32}\! \left(x \right) x +2 F_{32}\! \left(x \right)-1\\
F_{33}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= x^{2} F_{37} \left(x \right)^{2}+2 x^{2} F_{37}\! \left(x \right)-2 x F_{37} \left(x \right)^{2}+x^{2}-3 x F_{37}\! \left(x \right)-x +2 F_{37}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{15}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{68}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{15}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{79}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{40}\! \left(x \right)\\
\end{align*}\)