Av(12453, 12534, 12543, 13452, 13542, 14352, 21453, 21534, 21543, 31524)
Counting Sequence
1, 1, 2, 6, 24, 110, 533, 2673, 13757, 72266, 385940, 2089319, 11439790, 63242238, 352515817, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x^{6}-5 x^{5}+3 x^{4}-3 x^{3}+6 x^{2}-2 x -2\right) F \left(x
\right)^{3}-x \left(x^{6}-7 x^{5}+10 x^{4}-6 x^{3}+8 x^{2}-7 x -2\right) F \left(x
\right)^{2}-\left(x^{2}-2 x -1\right) \left(2 x^{4}-3 x^{3}+2 x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{2}+1\right) \left(x^{2}-x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 110\)
\(\displaystyle a \! \left(6\right) = 533\)
\(\displaystyle a \! \left(7\right) = 2673\)
\(\displaystyle a \! \left(8\right) = 13757\)
\(\displaystyle a \! \left(9\right) = 72266\)
\(\displaystyle a \! \left(10\right) = 385940\)
\(\displaystyle a \! \left(11\right) = 2089319\)
\(\displaystyle a \! \left(12\right) = 11439790\)
\(\displaystyle a \! \left(13\right) = 63242238\)
\(\displaystyle a \! \left(14\right) = 352515817\)
\(\displaystyle a \! \left(15\right) = 1979046864\)
\(\displaystyle a \! \left(16\right) = 11180267860\)
\(\displaystyle a \! \left(17\right) = 63510804376\)
\(\displaystyle a \! \left(18\right) = 362556716475\)
\(\displaystyle a \! \left(19\right) = 2078800648342\)
\(\displaystyle a \! \left(20\right) = 11966509335141\)
\(\displaystyle a \! \left(21\right) = 69131550779070\)
\(\displaystyle a \! \left(22\right) = 400680304821635\)
\(\displaystyle a \! \left(23\right) = 2329215624325159\)
\(\displaystyle a \! \left(24\right) = 13577001634509322\)
\(\displaystyle a \! \left(25\right) = 79338848765790520\)
\(\displaystyle a \! \left(26\right) = 464699339792308730\)
\(\displaystyle a \! \left(27\right) = 2727645599533038445\)
\(\displaystyle a \! \left(28\right) = 16042317812182880276\)
\(\displaystyle a \! \left(29\right) = 94525723736067575572\)
\(\displaystyle a \! \left(30\right) = 557934247481359819342\)
\(\displaystyle a \! \left(31\right) = 3298509385545323968801\)
\(\displaystyle a \! \left(32\right) = 19530359779784586127068\)
\(\displaystyle a \! \left(33\right) = 115803234634851747569433\)
\(\displaystyle a \! \left(34\right) = 687563313358980179596077\)
\(\displaystyle a \! \left(35\right) = 4087455241866433524477110\)
\(\displaystyle a \! \left(36\right) = 24328265974906652484037972\)
\(\displaystyle a \! \left(37\right) = 144963653160116788865119184\)
\(\displaystyle a \! \left(38\right) = 864711281402797301024675382\)
\(\displaystyle a \! \left(39\right) = 5163252287411397456977530156\)
\(\displaystyle a \! \left(40\right) = 30859846767054265447391352218\)
\(\displaystyle a \! \left(41\right) = 184612869011114357797882139173\)
\(\displaystyle a \! \left(42\right) = 1105373408381709459056462285334\)
\(\displaystyle a \! \left(43\right) = 6623953254928199696108353311228\)
\(\displaystyle a \! \left(44\right) = 39725583356912922419593221620088\)
\(\displaystyle a \! \left(45\right) = 238425528754059929098711919185366\)
\(\displaystyle a \! \left(46\right) = 1432024146163412587485686431312899\)
\(\displaystyle a \! \left(47\right) = 8606956871572391545886008173637700\)
\(\displaystyle a \! \left(48\right) = 51765216509953360509430420679528454\)
\(\displaystyle a \! \left(49\right) = 311532785426298198235672461809515279\)
\(\displaystyle a \! \left(50\right) = 1876012374278334412001617744332431882\)
\(\displaystyle a \! \left(n +51\right) = -\frac{n \left(n +2\right) a \! \left(n \right)}{8 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(5 n^{2}+15 n +8\right) a \! \left(n +1\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(23 n^{2}+187 n +300\right) a \! \left(n +2\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(73 n^{2}+515 n +854\right) a \! \left(n +3\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(145 n^{2}+1097 n +1866\right) a \! \left(n +4\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(179 n^{2}+2153 n +6228\right) a \! \left(n +5\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(247 n^{2}+3791 n +13340\right) a \! \left(n +6\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(997 n^{2}+17747 n +79050\right) a \! \left(n +7\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(558 n^{2}+12583 n +64690\right) a \! \left(n +8\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(85 n^{2}-5119 n -30054\right) a \! \left(n +9\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(12083 n^{2}+316537 n +1988244\right) a \! \left(n +10\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(6688 n^{2}+197387 n +1352133\right) a \! \left(n +11\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(12295 n^{2}+360533 n +2662716\right) a \! \left(n +12\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(7793 n^{2}+274009 n +2274138\right) a \! \left(n +13\right)}{16 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(1659 n^{2}+99431 n +1046436\right) a \! \left(n +14\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(10781 n^{2}+441019 n +4258431\right) a \! \left(n +15\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{9 \left(3037 n^{2}+168125 n +1926284\right) a \! \left(n +16\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(109609 n^{2}+4388381 n +43299096\right) a \! \left(n +17\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(90257 n^{2}+4094029 n +44645250\right) a \! \left(n +18\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(2986 n^{2}+98846 n +880347\right) a \! \left(n +19\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(226889 n^{2}+11515669 n +140767950\right) a \! \left(n +20\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(172919 n^{2}+9639985 n +126774906\right) a \! \left(n +21\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(4309 n^{2}+1295963 n +28147590\right) a \! \left(n +22\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(260 n^{2}-1068371 n -25544958\right) a \! \left(n +23\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(66519 n^{2}+2930639 n +31859810\right) a \! \left(n +24\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(95073 n^{2}+5570874 n +80157814\right) a \! \left(n +25\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(20473 n^{2}+182610 n -9298656\right) a \! \left(n +26\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(964085 n^{2}+55422007 n +791865954\right) a \! \left(n +27\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(618050 n^{2}+39390319 n +614370576\right) a \! \left(n +28\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(434522 n^{2}+25404124 n +362341623\right) a \! \left(n +29\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(720095 n^{2}+29049727 n +203437320\right) a \! \left(n +30\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(1405165 n^{2}+72878183 n +901291428\right) a \! \left(n +31\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(217771 n^{2}+13983175 n +227562428\right) a \! \left(n +32\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(817067 n^{2}+47290723 n +663554133\right) a \! \left(n +33\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(383128 n^{2}+24895513 n +403550962\right) a \! \left(n +34\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(462367 n^{2}+35863043 n +693680724\right) a \! \left(n +35\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(1602805 n^{2}+137276159 n +2885843160\right) a \! \left(n +36\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(856475 n^{2}+68127418 n +1349007570\right) a \! \left(n +37\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(442612 n^{2}+32140590 n +578076831\right) a \! \left(n +38\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(309147 n^{2}+21192810 n +350625457\right) a \! \left(n +39\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(629567 n^{2}+55759939 n +1232986278\right) a \! \left(n +40\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(702529 n^{2}+58407083 n +1214979753\right) a \! \left(n +41\right)}{16 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(165957 n^{2}+14172356 n +302625615\right) a \! \left(n +42\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(158792 n^{2}+14301775 n +321741069\right) a \! \left(n +43\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(134461 n^{2}+11362763 n +239290914\right) a \! \left(n +44\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(13270 n^{2}+1186021 n +26485254\right) a \! \left(n +45\right)}{8 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(1080 n^{2}+98293 n +2230546\right) a \! \left(n +46\right)}{8 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(9055 n^{2}+839531 n +19455456\right) a \! \left(n +47\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(53 n^{2}+6886 n +209304\right) a \! \left(n +48\right)}{4 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(160 n^{2}+16004 n +400209\right) a \! \left(n +49\right)}{4 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(34 n^{2}+3431 n +86553\right) a \! \left(n +50\right)}{4 \left(2 n +103\right) \left(n +51\right)}, \quad n \geq 51\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 110\)
\(\displaystyle a \! \left(6\right) = 533\)
\(\displaystyle a \! \left(7\right) = 2673\)
\(\displaystyle a \! \left(8\right) = 13757\)
\(\displaystyle a \! \left(9\right) = 72266\)
\(\displaystyle a \! \left(10\right) = 385940\)
\(\displaystyle a \! \left(11\right) = 2089319\)
\(\displaystyle a \! \left(12\right) = 11439790\)
\(\displaystyle a \! \left(13\right) = 63242238\)
\(\displaystyle a \! \left(14\right) = 352515817\)
\(\displaystyle a \! \left(15\right) = 1979046864\)
\(\displaystyle a \! \left(16\right) = 11180267860\)
\(\displaystyle a \! \left(17\right) = 63510804376\)
\(\displaystyle a \! \left(18\right) = 362556716475\)
\(\displaystyle a \! \left(19\right) = 2078800648342\)
\(\displaystyle a \! \left(20\right) = 11966509335141\)
\(\displaystyle a \! \left(21\right) = 69131550779070\)
\(\displaystyle a \! \left(22\right) = 400680304821635\)
\(\displaystyle a \! \left(23\right) = 2329215624325159\)
\(\displaystyle a \! \left(24\right) = 13577001634509322\)
\(\displaystyle a \! \left(25\right) = 79338848765790520\)
\(\displaystyle a \! \left(26\right) = 464699339792308730\)
\(\displaystyle a \! \left(27\right) = 2727645599533038445\)
\(\displaystyle a \! \left(28\right) = 16042317812182880276\)
\(\displaystyle a \! \left(29\right) = 94525723736067575572\)
\(\displaystyle a \! \left(30\right) = 557934247481359819342\)
\(\displaystyle a \! \left(31\right) = 3298509385545323968801\)
\(\displaystyle a \! \left(32\right) = 19530359779784586127068\)
\(\displaystyle a \! \left(33\right) = 115803234634851747569433\)
\(\displaystyle a \! \left(34\right) = 687563313358980179596077\)
\(\displaystyle a \! \left(35\right) = 4087455241866433524477110\)
\(\displaystyle a \! \left(36\right) = 24328265974906652484037972\)
\(\displaystyle a \! \left(37\right) = 144963653160116788865119184\)
\(\displaystyle a \! \left(38\right) = 864711281402797301024675382\)
\(\displaystyle a \! \left(39\right) = 5163252287411397456977530156\)
\(\displaystyle a \! \left(40\right) = 30859846767054265447391352218\)
\(\displaystyle a \! \left(41\right) = 184612869011114357797882139173\)
\(\displaystyle a \! \left(42\right) = 1105373408381709459056462285334\)
\(\displaystyle a \! \left(43\right) = 6623953254928199696108353311228\)
\(\displaystyle a \! \left(44\right) = 39725583356912922419593221620088\)
\(\displaystyle a \! \left(45\right) = 238425528754059929098711919185366\)
\(\displaystyle a \! \left(46\right) = 1432024146163412587485686431312899\)
\(\displaystyle a \! \left(47\right) = 8606956871572391545886008173637700\)
\(\displaystyle a \! \left(48\right) = 51765216509953360509430420679528454\)
\(\displaystyle a \! \left(49\right) = 311532785426298198235672461809515279\)
\(\displaystyle a \! \left(50\right) = 1876012374278334412001617744332431882\)
\(\displaystyle a \! \left(n +51\right) = -\frac{n \left(n +2\right) a \! \left(n \right)}{8 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(5 n^{2}+15 n +8\right) a \! \left(n +1\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(23 n^{2}+187 n +300\right) a \! \left(n +2\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(73 n^{2}+515 n +854\right) a \! \left(n +3\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(145 n^{2}+1097 n +1866\right) a \! \left(n +4\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(179 n^{2}+2153 n +6228\right) a \! \left(n +5\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(247 n^{2}+3791 n +13340\right) a \! \left(n +6\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(997 n^{2}+17747 n +79050\right) a \! \left(n +7\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(558 n^{2}+12583 n +64690\right) a \! \left(n +8\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(85 n^{2}-5119 n -30054\right) a \! \left(n +9\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(12083 n^{2}+316537 n +1988244\right) a \! \left(n +10\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(6688 n^{2}+197387 n +1352133\right) a \! \left(n +11\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(12295 n^{2}+360533 n +2662716\right) a \! \left(n +12\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(7793 n^{2}+274009 n +2274138\right) a \! \left(n +13\right)}{16 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(1659 n^{2}+99431 n +1046436\right) a \! \left(n +14\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(10781 n^{2}+441019 n +4258431\right) a \! \left(n +15\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{9 \left(3037 n^{2}+168125 n +1926284\right) a \! \left(n +16\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(109609 n^{2}+4388381 n +43299096\right) a \! \left(n +17\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(90257 n^{2}+4094029 n +44645250\right) a \! \left(n +18\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(2986 n^{2}+98846 n +880347\right) a \! \left(n +19\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(226889 n^{2}+11515669 n +140767950\right) a \! \left(n +20\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(172919 n^{2}+9639985 n +126774906\right) a \! \left(n +21\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(4309 n^{2}+1295963 n +28147590\right) a \! \left(n +22\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(260 n^{2}-1068371 n -25544958\right) a \! \left(n +23\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(66519 n^{2}+2930639 n +31859810\right) a \! \left(n +24\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(95073 n^{2}+5570874 n +80157814\right) a \! \left(n +25\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(20473 n^{2}+182610 n -9298656\right) a \! \left(n +26\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(964085 n^{2}+55422007 n +791865954\right) a \! \left(n +27\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(618050 n^{2}+39390319 n +614370576\right) a \! \left(n +28\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(434522 n^{2}+25404124 n +362341623\right) a \! \left(n +29\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(720095 n^{2}+29049727 n +203437320\right) a \! \left(n +30\right)}{64 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(1405165 n^{2}+72878183 n +901291428\right) a \! \left(n +31\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(217771 n^{2}+13983175 n +227562428\right) a \! \left(n +32\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(817067 n^{2}+47290723 n +663554133\right) a \! \left(n +33\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(383128 n^{2}+24895513 n +403550962\right) a \! \left(n +34\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(462367 n^{2}+35863043 n +693680724\right) a \! \left(n +35\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(1602805 n^{2}+137276159 n +2885843160\right) a \! \left(n +36\right)}{64 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(856475 n^{2}+68127418 n +1349007570\right) a \! \left(n +37\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(442612 n^{2}+32140590 n +578076831\right) a \! \left(n +38\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(309147 n^{2}+21192810 n +350625457\right) a \! \left(n +39\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(629567 n^{2}+55759939 n +1232986278\right) a \! \left(n +40\right)}{32 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(702529 n^{2}+58407083 n +1214979753\right) a \! \left(n +41\right)}{16 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(165957 n^{2}+14172356 n +302625615\right) a \! \left(n +42\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(158792 n^{2}+14301775 n +321741069\right) a \! \left(n +43\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(134461 n^{2}+11362763 n +239290914\right) a \! \left(n +44\right)}{32 \left(2 n +103\right) \left(n +51\right)}+\frac{3 \left(13270 n^{2}+1186021 n +26485254\right) a \! \left(n +45\right)}{8 \left(2 n +103\right) \left(n +51\right)}-\frac{3 \left(1080 n^{2}+98293 n +2230546\right) a \! \left(n +46\right)}{8 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(9055 n^{2}+839531 n +19455456\right) a \! \left(n +47\right)}{16 \left(2 n +103\right) \left(n +51\right)}-\frac{\left(53 n^{2}+6886 n +209304\right) a \! \left(n +48\right)}{4 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(160 n^{2}+16004 n +400209\right) a \! \left(n +49\right)}{4 \left(2 n +103\right) \left(n +51\right)}+\frac{\left(34 n^{2}+3431 n +86553\right) a \! \left(n +50\right)}{4 \left(2 n +103\right) \left(n +51\right)}, \quad n \geq 51\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 128 rules.
Found on January 25, 2022.Finding the specification took 4394 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_{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_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= -F_{107}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{101}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= -F_{37}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= -F_{42}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{11}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{11}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{11}\! \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) F_{51}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{11}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 0\\
F_{67}\! \left(x \right) &= F_{11}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{11}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 2 F_{66}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{11}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{11}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{11}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{66}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{11}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{88}\! \left(x \right) &= 2 F_{66}\! \left(x \right)+F_{75}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{93}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= -F_{95}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{11}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{0}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{11}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{0}\! \left(x \right) F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{45}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{0}\! \left(x \right) F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= -F_{119}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{11}\! \left(x \right) F_{116}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{24}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{24}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{0}\! \left(x \right) F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{2}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{125}\! \left(x \right) &= \frac{F_{126}\! \left(x \right)}{F_{46}\! \left(x \right)}\\
F_{126}\! \left(x \right) &= -F_{127}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{0}\! \left(x \right) F_{23}\! \left(x \right)\\
\end{align*}\)