Av(1342, 2413, 3124, 3142)
View Raw Data
Generating Function
\(\displaystyle -\frac{\left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(x^{4}-4 x^{3}+10 x^{2}-6 x +1\right)}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 20, 68, 231, 781, 2629, 8821, 29530, 98706, 329592, 1099792, 3668127, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{4}-4 x^{3}+10 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right)^{2} = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+6 a \! \left(n +3\right)+1, \quad n \geq 5\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{\left(\left(\left(\left(-172 \sqrt{3}-92 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(1720 \sqrt{3}-680 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-2000 \sqrt{43}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(5895 \sqrt{43}\, \sqrt{3}-17415\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+36000 \sqrt{3}\, \sqrt{43}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+4\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(-15480 \sqrt{3}-8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-154800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right)\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(-\frac{\sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(18 \sqrt{43}\, \sqrt{3}-786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}-76800}}{240}+\frac{\left(-131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}+\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{\left(\left(\left(\left(172 \sqrt{3}+92 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-1720 \sqrt{3}+680 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+2000 \sqrt{43}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(-5895 \sqrt{43}\, \sqrt{3}+17415\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-36000 \sqrt{3}\, \sqrt{43}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+4\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(-15480 \sqrt{3}-8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-154800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right)\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(\frac{\sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(18 \sqrt{43}\, \sqrt{3}-786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}-76800}}{240}+\frac{\left(-131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}+\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{\left(\left(\left(\left(42 \,\mathrm{I} \sqrt{3}\, \sqrt{43}-1834 \,\mathrm{I}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-11200 \,\mathrm{I} \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-94000 \,\mathrm{I}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(100215 \,\mathrm{I} \sqrt{3}-6885 \,\mathrm{I} \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+612000 \,\mathrm{I} \sqrt{3}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{92}{17}\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(15480 \sqrt{3}+8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+154800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right)\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(-\frac{\mathrm{I} \sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(-18 \sqrt{43}\, \sqrt{3}+786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+76800}}{240}+\frac{\left(131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}-\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{1}{2}+\frac{\left(\left(\left(\left(-42 \,\mathrm{I} \sqrt{3}\, \sqrt{43}+1834 \,\mathrm{I}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+11200 \,\mathrm{I} \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+94000 \,\mathrm{I}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(-100215 \,\mathrm{I} \sqrt{3}+6885 \,\mathrm{I} \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-612000 \,\mathrm{I} \sqrt{3}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{92}{17}\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(15480 \sqrt{3}+8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+154800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right)\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(\frac{\mathrm{I} \sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(-18 \sqrt{43}\, \sqrt{3}+786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+76800}}{240}+\frac{\left(131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}-\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 40 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 40 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_{16}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{35}\! \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_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\ \end{align*}\)