Av(12453, 12534, 12543, 13452, 13524, 13542, 14523, 14532, 21453, 21534, 21543)
Counting Sequence
1, 1, 2, 6, 24, 109, 527, 2651, 13722, 72599, 390840, 2134152, 11791460, 65800151, 370320394, ...
Implicit Equation for the Generating Function
\(\displaystyle 8 x \left(x -1\right)^{2} F \left(x
\right)^{4}-2 x \left(x -1\right) \left(5 x -9\right) F \left(x
\right)^{3}+\left(5 x^{3}-19 x^{2}+18 x -2\right) F \left(x
\right)^{2}+\left(-x^{3}+4 x^{2}-7 x +3\right) F \! \left(x \right)-1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 109\)
\(\displaystyle a \! \left(6\right) = 527\)
\(\displaystyle a \! \left(7\right) = 2651\)
\(\displaystyle a \! \left(8\right) = 13722\)
\(\displaystyle a \! \left(9\right) = 72599\)
\(\displaystyle a \! \left(10\right) = 390840\)
\(\displaystyle a \! \left(11\right) = 2134152\)
\(\displaystyle a \! \left(12\right) = 11791460\)
\(\displaystyle a \! \left(13\right) = 65800151\)
\(\displaystyle a \! \left(14\right) = 370320394\)
\(\displaystyle a \! \left(15\right) = 2099523751\)
\(\displaystyle a \! \left(16\right) = 11979868741\)
\(\displaystyle a \! \left(17\right) = 68744714334\)
\(\displaystyle a \! \left(18\right) = 396466427341\)
\(\displaystyle a \! \left(19\right) = 2296790216854\)
\(\displaystyle a \! \left(20\right) = 13359406955245\)
\(\displaystyle a \! \left(21\right) = 77989387491127\)
\(\displaystyle a \! \left(22\right) = 456794701992937\)
\(\displaystyle a \! \left(23\right) = 2683599419586272\)
\(\displaystyle a \! \left(24\right) = 15809392256610621\)
\(\displaystyle a \! \left(25\right) = 93371993212349838\)
\(\displaystyle a \! \left(26\right) = 552759784817589174\)
\(\displaystyle a \! \left(27\right) = 3279431602615205345\)
\(\displaystyle a \! \left(28\right) = 19495538137352113600\)
\(\displaystyle a \! \left(29\right) = 116114281590001668243\)
\(\displaystyle a \! \left(30\right) = 692779745366901449978\)
\(\displaystyle a \! \left(31\right) = 4140135418041075187158\)
\(\displaystyle a \! \left(32\right) = 24779873157657253084662\)
\(\displaystyle a \! \left(n +33\right) = \frac{\left(5453129 n^{3}+58746030 n^{2}+212610403 n +256416318\right) a \! \left(n +3\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(46562725 n^{3}+641128209 n^{2}+2957288774 n +4552192776\right) a \! \left(n +4\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(326589125 n^{3}+5476297116 n^{2}+30608587741 n +56958242694\right) a \! \left(n +5\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(1871737283 n^{3}+36759607065 n^{2}+239903815120 n +520277329572\right) a \! \left(n +6\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(8566095253 n^{3}+191571077352 n^{2}+1421787323753 n +3502746080298\right) a \! \left(n +7\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(30579736123 n^{3}+763792246539 n^{2}+6326243040410 n +17378509264704\right) a \! \left(n +8\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(83633404261 n^{3}+2297717916618 n^{2}+20912257056503 n +63041671597458\right) a \! \left(n +9\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(171341752999 n^{3}+5096234779749 n^{2}+50099115647888 n +162644214824580\right) a \! \left(n +10\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(83171698037 n^{3}+2610363904468 n^{2}+26880564100817 n +90484486516790\right) a \! \left(n +11\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(105698513102 n^{3}+3184497191289 n^{2}+29708226424000 n +80443208898264\right) a \! \left(n +12\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{1561 n \left(n +1\right) \left(n +2\right) a \! \left(n \right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(33707 n +82869\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(n +2\right) \left(249923 n^{2}+1479200 n +2261154\right) a \! \left(n +2\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(13985498639 n^{3}+1193862380010 n^{2}+24802348571488 n +151497176604645\right) a \! \left(n +13\right)}{6528 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(188876689495 n^{3}+9323009650707 n^{2}+151513812321192 n +812342000243852\right) a \! \left(n +14\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(589934531224 n^{3}+29253725282169 n^{2}+482013418253171 n +2639450626523388\right) a \! \left(n +15\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(1660557625301 n^{3}+85488426467451 n^{2}+1465337592325486 n +8362612214475696\right) a \! \left(n +16\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{3 \left(202662656877 n^{3}+10904180956320 n^{2}+195487480494185 n +1167717949286726\right) a \! \left(n +17\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(825028569164 n^{3}+46436274419793 n^{2}+871220965626172 n +5448401732037924\right) a \! \left(n +18\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(423564009957 n^{3}+24925362196102 n^{2}+489080027186287 n +3199812496039262\right) a \! \left(n +19\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(142889027964 n^{3}+8786254251751 n^{2}+180200944684872 n +1232685281770184\right) a \! \left(n +20\right)}{4352 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(260903696114 n^{3}+16759586252313 n^{2}+359193265112989 n +2568451614673362\right) a \! \left(n +21\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(293864113213 n^{3}+19716936815961 n^{2}+441483826468352 n +3298912238615220\right) a \! \left(n +22\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(77214909596 n^{3}+5408903662425 n^{2}+126451405573477 n +986602925993142\right) a \! \left(n +23\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(37328938249 n^{3}+2727679754997 n^{2}+66508831297022 n +541128281320896\right) a \! \left(n +24\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(16314123629 n^{3}+1241612117205 n^{2}+31521460376008 n +266944498898376\right) a \! \left(n +25\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(1081658959 n^{3}+85500599644 n^{2}+2253613076055 n +19806949536878\right) a \! \left(n +26\right)}{2176 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(1205464037 n^{3}+98592336531 n^{2}+2687987234320 n +24428883515748\right) a \! \left(n +27\right)}{6528 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(207146209 n^{3}+17483553684 n^{2}+491814148073 n +4610922749772\right) a \! \left(n +28\right)}{3264 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(60914161 n^{3}+5303314029 n^{2}+153874592564 n +1487900593428\right) a \! \left(n +29\right)}{3264 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(1139267 n^{3}+102343929 n^{2}+3063985994 n +30570119888\right) a \! \left(n +30\right)}{272 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(344695 n^{3}+31944843 n^{2}+986645662 n +10155812224\right) a \! \left(n +31\right)}{544 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(7475 n^{2}+475033 n +7543108\right) a \! \left(n +32\right)}{136 \left(2 n +67\right) \left(n +33\right)}, \quad n \geq 33\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 109\)
\(\displaystyle a \! \left(6\right) = 527\)
\(\displaystyle a \! \left(7\right) = 2651\)
\(\displaystyle a \! \left(8\right) = 13722\)
\(\displaystyle a \! \left(9\right) = 72599\)
\(\displaystyle a \! \left(10\right) = 390840\)
\(\displaystyle a \! \left(11\right) = 2134152\)
\(\displaystyle a \! \left(12\right) = 11791460\)
\(\displaystyle a \! \left(13\right) = 65800151\)
\(\displaystyle a \! \left(14\right) = 370320394\)
\(\displaystyle a \! \left(15\right) = 2099523751\)
\(\displaystyle a \! \left(16\right) = 11979868741\)
\(\displaystyle a \! \left(17\right) = 68744714334\)
\(\displaystyle a \! \left(18\right) = 396466427341\)
\(\displaystyle a \! \left(19\right) = 2296790216854\)
\(\displaystyle a \! \left(20\right) = 13359406955245\)
\(\displaystyle a \! \left(21\right) = 77989387491127\)
\(\displaystyle a \! \left(22\right) = 456794701992937\)
\(\displaystyle a \! \left(23\right) = 2683599419586272\)
\(\displaystyle a \! \left(24\right) = 15809392256610621\)
\(\displaystyle a \! \left(25\right) = 93371993212349838\)
\(\displaystyle a \! \left(26\right) = 552759784817589174\)
\(\displaystyle a \! \left(27\right) = 3279431602615205345\)
\(\displaystyle a \! \left(28\right) = 19495538137352113600\)
\(\displaystyle a \! \left(29\right) = 116114281590001668243\)
\(\displaystyle a \! \left(30\right) = 692779745366901449978\)
\(\displaystyle a \! \left(31\right) = 4140135418041075187158\)
\(\displaystyle a \! \left(32\right) = 24779873157657253084662\)
\(\displaystyle a \! \left(n +33\right) = \frac{\left(5453129 n^{3}+58746030 n^{2}+212610403 n +256416318\right) a \! \left(n +3\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(46562725 n^{3}+641128209 n^{2}+2957288774 n +4552192776\right) a \! \left(n +4\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(326589125 n^{3}+5476297116 n^{2}+30608587741 n +56958242694\right) a \! \left(n +5\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(1871737283 n^{3}+36759607065 n^{2}+239903815120 n +520277329572\right) a \! \left(n +6\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(8566095253 n^{3}+191571077352 n^{2}+1421787323753 n +3502746080298\right) a \! \left(n +7\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(30579736123 n^{3}+763792246539 n^{2}+6326243040410 n +17378509264704\right) a \! \left(n +8\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(83633404261 n^{3}+2297717916618 n^{2}+20912257056503 n +63041671597458\right) a \! \left(n +9\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(171341752999 n^{3}+5096234779749 n^{2}+50099115647888 n +162644214824580\right) a \! \left(n +10\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(83171698037 n^{3}+2610363904468 n^{2}+26880564100817 n +90484486516790\right) a \! \left(n +11\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(105698513102 n^{3}+3184497191289 n^{2}+29708226424000 n +80443208898264\right) a \! \left(n +12\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{1561 n \left(n +1\right) \left(n +2\right) a \! \left(n \right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(33707 n +82869\right) \left(n +2\right) \left(n +1\right) a \! \left(n +1\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(n +2\right) \left(249923 n^{2}+1479200 n +2261154\right) a \! \left(n +2\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(13985498639 n^{3}+1193862380010 n^{2}+24802348571488 n +151497176604645\right) a \! \left(n +13\right)}{6528 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(188876689495 n^{3}+9323009650707 n^{2}+151513812321192 n +812342000243852\right) a \! \left(n +14\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(589934531224 n^{3}+29253725282169 n^{2}+482013418253171 n +2639450626523388\right) a \! \left(n +15\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(1660557625301 n^{3}+85488426467451 n^{2}+1465337592325486 n +8362612214475696\right) a \! \left(n +16\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{3 \left(202662656877 n^{3}+10904180956320 n^{2}+195487480494185 n +1167717949286726\right) a \! \left(n +17\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(825028569164 n^{3}+46436274419793 n^{2}+871220965626172 n +5448401732037924\right) a \! \left(n +18\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(423564009957 n^{3}+24925362196102 n^{2}+489080027186287 n +3199812496039262\right) a \! \left(n +19\right)}{8704 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(142889027964 n^{3}+8786254251751 n^{2}+180200944684872 n +1232685281770184\right) a \! \left(n +20\right)}{4352 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(260903696114 n^{3}+16759586252313 n^{2}+359193265112989 n +2568451614673362\right) a \! \left(n +21\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(293864113213 n^{3}+19716936815961 n^{2}+441483826468352 n +3298912238615220\right) a \! \left(n +22\right)}{26112 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(77214909596 n^{3}+5408903662425 n^{2}+126451405573477 n +986602925993142\right) a \! \left(n +23\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(37328938249 n^{3}+2727679754997 n^{2}+66508831297022 n +541128281320896\right) a \! \left(n +24\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(16314123629 n^{3}+1241612117205 n^{2}+31521460376008 n +266944498898376\right) a \! \left(n +25\right)}{13056 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(1081658959 n^{3}+85500599644 n^{2}+2253613076055 n +19806949536878\right) a \! \left(n +26\right)}{2176 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(1205464037 n^{3}+98592336531 n^{2}+2687987234320 n +24428883515748\right) a \! \left(n +27\right)}{6528 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(207146209 n^{3}+17483553684 n^{2}+491814148073 n +4610922749772\right) a \! \left(n +28\right)}{3264 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(60914161 n^{3}+5303314029 n^{2}+153874592564 n +1487900593428\right) a \! \left(n +29\right)}{3264 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(1139267 n^{3}+102343929 n^{2}+3063985994 n +30570119888\right) a \! \left(n +30\right)}{272 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}-\frac{\left(344695 n^{3}+31944843 n^{2}+986645662 n +10155812224\right) a \! \left(n +31\right)}{544 \left(n +33\right) \left(n +32\right) \left(2 n +67\right)}+\frac{\left(7475 n^{2}+475033 n +7543108\right) a \! \left(n +32\right)}{136 \left(2 n +67\right) \left(n +33\right)}, \quad n \geq 33\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 154 rules.
Found on January 25, 2022.Finding the specification took 1947 seconds.
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Copy 154 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_{11}\! \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_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= \frac{F_{10}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= -F_{38}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \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_{11}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{0}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{26}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{11}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{139}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{11}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{15}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{64}\! \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_{11}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{11}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= 0\\
F_{69}\! \left(x \right) &= F_{11}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{11}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{74}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{74}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{11} \left(x \right)^{2} F_{66}\! \left(x \right)}\\
F_{75}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{11}\! \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_{1}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{11}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{11}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= x^{2}\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{11}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{11}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{101}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{102}\! \left(x \right) &= 2 F_{68}\! \left(x \right)+F_{103}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{11}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{115}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{115}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{11}\! \left(x \right) F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{11}\! \left(x \right) F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{119}\! \left(x \right) &= 2 F_{68}\! \left(x \right)+F_{120}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{11}\! \left(x \right) F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{11}\! \left(x \right) F_{112}\! \left(x \right)\\
F_{123}\! \left(x \right) &= 2 F_{68}\! \left(x \right)+F_{124}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{11}\! \left(x \right) F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{128}\! \left(x \right) &= 2 F_{68}\! \left(x \right)+F_{129}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{11}\! \left(x \right) F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{11}\! \left(x \right) F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= 3 F_{68}\! \left(x \right)+F_{136}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{11}\! \left(x \right) F_{137}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{11}\! \left(x \right) F_{128}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{11}\! \left(x \right) F_{141}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{11}\! \left(x \right) F_{145}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{148}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{152}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{11}\! \left(x \right) F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{11}\! \left(x \right) F_{153}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{64}\! \left(x \right)\\
\end{align*}\)