Av(14352, 14523, 14532, 41352, 41523, 41532, 45123, 45132)
Counting Sequence
1, 1, 2, 6, 24, 112, 562, 2927, 15608, 84718, 466543, 2600700, 14648080, 83238485, 476651891, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(29 x^{2}-30 x +9\right) \left(x -1\right)^{3} F \left(x
\right)^{3}+x \left(x^{4}+6 x^{3}+29 x^{2}-36 x +12\right) \left(x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(x^{6}-11 x^{5}+35 x^{4}-61 x^{3}+37 x^{2}-5 x -2\right) F \! \left(x \right)+2 x^{6}-7 x^{5}+10 x^{4}-9 x^{3}+10 x^{2}-7 x +2 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 562\)
\(\displaystyle a(7) = 2927\)
\(\displaystyle a(8) = 15608\)
\(\displaystyle a(9) = 84718\)
\(\displaystyle a(10) = 466543\)
\(\displaystyle a(11) = 2600700\)
\(\displaystyle a(12) = 14648080\)
\(\displaystyle a(13) = 83238485\)
\(\displaystyle a(14) = 476651891\)
\(\displaystyle a(15) = 2747815927\)
\(\displaystyle a(16) = 15934310868\)
\(\displaystyle a(17) = 92885358471\)
\(\displaystyle a{\left(n + 18 \right)} = - \frac{1370395 n \left(n + 1\right) a{\left(n \right)}}{12096 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(n + 1\right) \left(19748351 n + 32407848\right) a{\left(n + 1 \right)}}{3024 \left(n + 18\right) \left(n + 19\right)} + \frac{19 \left(275 n + 4694\right) a{\left(n + 17 \right)}}{168 \left(n + 19\right)} - \frac{\left(896398 n^{2} + 29681593 n + 245526552\right) a{\left(n + 16 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(15750767 n^{2} + 490443785 n + 3814828188\right) a{\left(n + 15 \right)}}{4032 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(143492824 n^{2} + 4182864997 n + 30455827494\right) a{\left(n + 14 \right)}}{6048 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(160313728 n^{2} + 4352875423 n + 29516733636\right) a{\left(n + 13 \right)}}{1512 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(411978649 n^{2} + 1959979996 n + 2288866464\right) a{\left(n + 2 \right)}}{6048 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(1402834293 n^{2} + 9615030827 n + 16305369592\right) a{\left(n + 3 \right)}}{4032 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(2223639108 n^{2} + 19841243905 n + 43938182516\right) a{\left(n + 4 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(3914330974 n^{2} + 50924285413 n + 164897460121\right) a{\left(n + 6 \right)}}{1008 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(4375399333 n^{2} + 110021209585 n + 690766016952\right) a{\left(n + 12 \right)}}{12096 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(4786568601 n^{2} + 71991079797 n + 269732343946\right) a{\left(n + 7 \right)}}{1008 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(4871304425 n^{2} + 53449435069 n + 145799584556\right) a{\left(n + 5 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(5826453809 n^{2} + 134776308413 n + 778215019578\right) a{\left(n + 11 \right)}}{6048 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(6134945095 n^{2} + 129528488683 n + 682408932162\right) a{\left(n + 10 \right)}}{3024 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(9113846741 n^{2} + 155545731093 n + 661776041168\right) a{\left(n + 8 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(10273144583 n^{2} + 196127651735 n + 933916986084\right) a{\left(n + 9 \right)}}{3024 \left(n + 18\right) \left(n + 19\right)} - \frac{10}{63 \left(n + 18\right) \left(n + 19\right)}, \quad n \geq 18\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 562\)
\(\displaystyle a(7) = 2927\)
\(\displaystyle a(8) = 15608\)
\(\displaystyle a(9) = 84718\)
\(\displaystyle a(10) = 466543\)
\(\displaystyle a(11) = 2600700\)
\(\displaystyle a(12) = 14648080\)
\(\displaystyle a(13) = 83238485\)
\(\displaystyle a(14) = 476651891\)
\(\displaystyle a(15) = 2747815927\)
\(\displaystyle a(16) = 15934310868\)
\(\displaystyle a(17) = 92885358471\)
\(\displaystyle a{\left(n + 18 \right)} = - \frac{1370395 n \left(n + 1\right) a{\left(n \right)}}{12096 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(n + 1\right) \left(19748351 n + 32407848\right) a{\left(n + 1 \right)}}{3024 \left(n + 18\right) \left(n + 19\right)} + \frac{19 \left(275 n + 4694\right) a{\left(n + 17 \right)}}{168 \left(n + 19\right)} - \frac{\left(896398 n^{2} + 29681593 n + 245526552\right) a{\left(n + 16 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(15750767 n^{2} + 490443785 n + 3814828188\right) a{\left(n + 15 \right)}}{4032 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(143492824 n^{2} + 4182864997 n + 30455827494\right) a{\left(n + 14 \right)}}{6048 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(160313728 n^{2} + 4352875423 n + 29516733636\right) a{\left(n + 13 \right)}}{1512 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(411978649 n^{2} + 1959979996 n + 2288866464\right) a{\left(n + 2 \right)}}{6048 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(1402834293 n^{2} + 9615030827 n + 16305369592\right) a{\left(n + 3 \right)}}{4032 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(2223639108 n^{2} + 19841243905 n + 43938182516\right) a{\left(n + 4 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(3914330974 n^{2} + 50924285413 n + 164897460121\right) a{\left(n + 6 \right)}}{1008 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(4375399333 n^{2} + 110021209585 n + 690766016952\right) a{\left(n + 12 \right)}}{12096 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(4786568601 n^{2} + 71991079797 n + 269732343946\right) a{\left(n + 7 \right)}}{1008 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(4871304425 n^{2} + 53449435069 n + 145799584556\right) a{\left(n + 5 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(5826453809 n^{2} + 134776308413 n + 778215019578\right) a{\left(n + 11 \right)}}{6048 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(6134945095 n^{2} + 129528488683 n + 682408932162\right) a{\left(n + 10 \right)}}{3024 \left(n + 18\right) \left(n + 19\right)} - \frac{\left(9113846741 n^{2} + 155545731093 n + 661776041168\right) a{\left(n + 8 \right)}}{2016 \left(n + 18\right) \left(n + 19\right)} + \frac{\left(10273144583 n^{2} + 196127651735 n + 933916986084\right) a{\left(n + 9 \right)}}{3024 \left(n + 18\right) \left(n + 19\right)} - \frac{10}{63 \left(n + 18\right) \left(n + 19\right)}, \quad n \geq 18\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 127 rules.
Finding the specification took 24439 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_{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_{33}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}-2 x F_{11} \left(x \right)^{2}+F_{11}\! \left(x \right) x +2 F_{11}\! \left(x \right)-1\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= -F_{11}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{0}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{44}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{0}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{44}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \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_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{93}\! \left(x \right)}\\
F_{67}\! \left(x \right) &= -F_{99}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{96}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{76}\! \left(x \right) &= \frac{F_{77}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{2}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{45}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{0}\! \left(x \right) F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{102}\! \left(x \right) &= -F_{113}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= \frac{F_{104}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{109}\! \left(x \right) &= \frac{F_{110}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{110}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{11}\! \left(x \right) F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{11}\! \left(x \right) F_{85}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 127 rules.
Finding the specification took 64327 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_{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_{33}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}-2 x F_{11} \left(x \right)^{2}+F_{11}\! \left(x \right) x +2 F_{11}\! \left(x \right)-1\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{6}\! \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_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{0}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{44}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{0}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{44}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \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_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{93}\! \left(x \right)}\\
F_{67}\! \left(x \right) &= -F_{99}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{96}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{76}\! \left(x \right) &= \frac{F_{77}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{2}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{45}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{0}\! \left(x \right) F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{102}\! \left(x \right) &= -F_{113}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= \frac{F_{104}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{109}\! \left(x \right) &= \frac{F_{110}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{110}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{11}\! \left(x \right) F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{11}\! \left(x \right) F_{85}\! \left(x \right)\\
\end{align*}\)