Av(13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432)
Counting Sequence
1, 1, 2, 6, 24, 112, 568, 3034, 16804, 95612, 555492, 3281540, 19651112, 119018910, 727787376, ...
Implicit Equation for the Generating Function
\(\displaystyle F \left(x
\right)^{4}+\left(-2 x -2\right) F \left(x
\right)^{3}+\left(x^{2}+6 x -1\right) F \left(x
\right)^{2}+\left(-6 x +4\right) F \! \left(x \right)+2 x -2 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 568\)
\(\displaystyle a(7) = 3034\)
\(\displaystyle a(8) = 16804\)
\(\displaystyle a(9) = 95612\)
\(\displaystyle a(10) = 555492\)
\(\displaystyle a(11) = 3281540\)
\(\displaystyle a(12) = 19651112\)
\(\displaystyle a{\left(n + 13 \right)} = - \frac{48 \left(n - 1\right) \left(2 n - 1\right) \left(2 n + 1\right) a{\left(n \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{\left(182 n + 1853\right) a{\left(n + 12 \right)}}{86 \left(n + 13\right)} + \frac{\left(5951 n^{2} + 125222 n + 660732\right) a{\left(n + 11 \right)}}{86 \left(n + 12\right) \left(n + 13\right)} - \frac{12 \left(2 n + 1\right) \left(54 n^{2} - 75 n - 4\right) a{\left(n + 1 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{12 \left(970 n^{3} + 2735 n^{2} + 1988 n + 242\right) a{\left(n + 2 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{6 \left(7240 n^{3} + 45110 n^{2} + 93311 n + 65802\right) a{\left(n + 3 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{10 \left(12752 n^{3} + 129726 n^{2} + 442825 n + 509838\right) a{\left(n + 4 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{\left(19333 n^{3} + 450900 n^{2} + 3424580 n + 8400570\right) a{\left(n + 9 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{\left(20750 n^{3} + 611295 n^{2} + 6019213 n + 19810314\right) a{\left(n + 10 \right)}}{86 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{6 \left(46435 n^{3} + 641175 n^{2} + 2953697 n + 4543646\right) a{\left(n + 5 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{5 \left(66857 n^{3} + 1306128 n^{2} + 8483965 n + 18324186\right) a{\left(n + 7 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{\left(153211 n^{3} + 3370986 n^{2} + 24602993 n + 59545710\right) a{\left(n + 8 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{\left(392560 n^{3} + 6617835 n^{2} + 37135823 n + 69383538\right) a{\left(n + 6 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)}, \quad n \geq 13\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 568\)
\(\displaystyle a(7) = 3034\)
\(\displaystyle a(8) = 16804\)
\(\displaystyle a(9) = 95612\)
\(\displaystyle a(10) = 555492\)
\(\displaystyle a(11) = 3281540\)
\(\displaystyle a(12) = 19651112\)
\(\displaystyle a{\left(n + 13 \right)} = - \frac{48 \left(n - 1\right) \left(2 n - 1\right) \left(2 n + 1\right) a{\left(n \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{\left(182 n + 1853\right) a{\left(n + 12 \right)}}{86 \left(n + 13\right)} + \frac{\left(5951 n^{2} + 125222 n + 660732\right) a{\left(n + 11 \right)}}{86 \left(n + 12\right) \left(n + 13\right)} - \frac{12 \left(2 n + 1\right) \left(54 n^{2} - 75 n - 4\right) a{\left(n + 1 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{12 \left(970 n^{3} + 2735 n^{2} + 1988 n + 242\right) a{\left(n + 2 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{6 \left(7240 n^{3} + 45110 n^{2} + 93311 n + 65802\right) a{\left(n + 3 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{10 \left(12752 n^{3} + 129726 n^{2} + 442825 n + 509838\right) a{\left(n + 4 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{\left(19333 n^{3} + 450900 n^{2} + 3424580 n + 8400570\right) a{\left(n + 9 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{\left(20750 n^{3} + 611295 n^{2} + 6019213 n + 19810314\right) a{\left(n + 10 \right)}}{86 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{6 \left(46435 n^{3} + 641175 n^{2} + 2953697 n + 4543646\right) a{\left(n + 5 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} - \frac{5 \left(66857 n^{3} + 1306128 n^{2} + 8483965 n + 18324186\right) a{\left(n + 7 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{\left(153211 n^{3} + 3370986 n^{2} + 24602993 n + 59545710\right) a{\left(n + 8 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)} + \frac{\left(392560 n^{3} + 6617835 n^{2} + 37135823 n + 69383538\right) a{\left(n + 6 \right)}}{43 \left(n + 11\right) \left(n + 12\right) \left(n + 13\right)}, \quad n \geq 13\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 112 rules.
Finding the specification took 13250 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_{22}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{22}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{88}\! \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_{27}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{22}\! \left(x \right) &= x\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{22}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{22}\! \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_{0}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{22}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{0}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{79}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= -F_{74}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= -F_{43}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{22}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{0}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= \frac{F_{53}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{17}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{22}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{22}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= -F_{73}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= -F_{72}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{22}\! \left(x \right) F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{17}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{0}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{22}\! \left(x \right) F_{36}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= \frac{F_{77}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= -F_{79}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{22}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{108}\! \left(x \right) F_{22}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{22}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{0}\! \left(x \right) F_{100}\! \left(x \right) F_{17}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= -F_{104}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= \frac{F_{103}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{103}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{106}\! \left(x \right) &= \frac{F_{107}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{107}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{108}\! \left(x \right) &= \frac{F_{109}\! \left(x \right)}{F_{22}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{109}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{100}\! \left(x \right) F_{108}\! \left(x \right) F_{22}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 97 rules.
Finding the specification took 38226 seconds.
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Copy 97 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_{68}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
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) &= F_{13}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{0}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{21}\! \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_{5}\! \left(x \right)\\
F_{21}\! \left(x \right) &= x\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{0}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{43}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= -F_{38}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{33}\! \left(x \right) &= -F_{37}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{21}\! \left(x \right) F_{26}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= -F_{43}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{21}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{19}\! \left(x \right) F_{21}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{21}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{21}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{67}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= -F_{16}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{21}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{74}\! \left(x \right) &= -F_{78}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= \frac{F_{76}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{21}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= \frac{F_{81}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{21}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{93}\! \left(x \right) &= \frac{F_{94}\! \left(x \right)}{F_{21}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{94}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{21}\! \left(x \right) F_{80}\! \left(x \right) F_{93}\! \left(x \right)\\
\end{align*}\)