Av(12453, 12534, 12543, 13452, 13542, 14352, 23451, 23541, 24351, 34251)
Counting Sequence
1, 1, 2, 6, 24, 110, 533, 2658, 13527, 69934, 366181, 1937809, 10348422, 55702889, 301933269, ...
Implicit Equation for the Generating Function
\(\displaystyle 22 x^{3} \left(x^{4}+5 x^{3}+7 x^{2}+3 x +1\right) \left(x -1\right)^{4} \left(2 x^{3}+2 x^{2}+2 x -1\right)^{4} F \left(x
\right)^{4}-x^{2} \left(66 x^{9}+220 x^{8}-116 x^{7}-1129 x^{6}-2413 x^{5}-2601 x^{4}-831 x^{3}+43 x^{2}+568 x -126\right) \left(x -1\right)^{3} \left(2 x^{3}+2 x^{2}+2 x -1\right)^{3} F \left(x
\right)^{3}-x \left(132 x^{13}+656 x^{12}+1528 x^{11}+2356 x^{10}-1480 x^{9}-10694 x^{8}-18523 x^{7}-14936 x^{6}-142 x^{5}+2514 x^{4}+2064 x^{3}-4164 x^{2}+2364 x -402\right) \left(x -1\right)^{2} \left(2 x^{3}+2 x^{2}+2 x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(2 x^{3}+2 x^{2}+2 x -1\right) \left(88 x^{17}+408 x^{16}+1120 x^{15}+4332 x^{14}+10776 x^{13}+12850 x^{12}-3510 x^{11}-33824 x^{10}-45457 x^{9}-23370 x^{8}+6372 x^{7}+6279 x^{6}-2512 x^{5}-4369 x^{4}+3761 x^{3}+413 x^{2}-780 x +134\right) F \! \left(x \right)+134+608 x^{19}+880 x^{18}+176 x^{20}-1450 x +5685 x^{2}-6410 x^{5}+6219 x^{4}-9407 x^{3}-24526 x^{13}-6548 x^{12}+22682 x^{11}+43747 x^{10}+8151 x^{9}-350 x^{8}-18776 x^{7}+16812 x^{6}+2360 x^{17}+2280 x^{16}-1840 x^{15}-15912 x^{14} = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 110\)
\(\displaystyle a(6) = 533\)
\(\displaystyle a(7) = 2658\)
\(\displaystyle a(8) = 13527\)
\(\displaystyle a(9) = 69934\)
\(\displaystyle a(10) = 366181\)
\(\displaystyle a(11) = 1937809\)
\(\displaystyle a(12) = 10348422\)
\(\displaystyle a(13) = 55702889\)
\(\displaystyle a(14) = 301933269\)
\(\displaystyle a(15) = 1646763624\)
\(\displaystyle a(16) = 9031253272\)
\(\displaystyle a(17) = 49775122555\)
\(\displaystyle a(18) = 275555847961\)
\(\displaystyle a(19) = 1531630492953\)
\(\displaystyle a(20) = 8544442259905\)
\(\displaystyle a(21) = 47825277966469\)
\(\displaystyle a(22) = 268504431166819\)
\(\displaystyle a(23) = 1511668242788459\)
\(\displaystyle a(24) = 8532497243252199\)
\(\displaystyle a(25) = 48275286710253950\)
\(\displaystyle a(26) = 273732484745996878\)
\(\displaystyle a(27) = 1555293598061318839\)
\(\displaystyle a(28) = 8853638333167138919\)
\(\displaystyle a(29) = 50489303711252247215\)
\(\displaystyle a(30) = 288400078142183511672\)
\(\displaystyle a(31) = 1649926475978732728306\)
\(\displaystyle a(32) = 9452919852620993040184\)
\(\displaystyle a(33) = 54232808019471321040944\)
\(\displaystyle a(34) = 311543517028359215062863\)
\(\displaystyle a(35) = 1791861729398854125336248\)
\(\displaystyle a(36) = 10317883180154667918146993\)
\(\displaystyle a(37) = 59477201630628447417042871\)
\(\displaystyle a(38) = 343209875284884106051289402\)
\(\displaystyle a(39) = 1982420205876416424049736851\)
\(\displaystyle a(40) = 11461392237015753534519534229\)
\(\displaystyle a(41) = 66323171892577862281148462095\)
\(\displaystyle a(42) = 384115075132653459782420361776\)
\(\displaystyle a(43) = 2226428261782638194179854356968\)
\(\displaystyle a{\left(n + 44 \right)} = - \frac{7170000 n \left(n + 1\right) \left(n + 2\right) a{\left(n \right)}}{106061 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{3000 \left(n + 1\right) \left(n + 2\right) \left(75444 n + 243221\right) a{\left(n + 1 \right)}}{106061 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{100 \left(n + 2\right) \left(92067333 n^{2} + 674308332 n + 1191810139\right) a{\left(n + 2 \right)}}{318183 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(11568193 n + 511623053\right) a{\left(n + 43 \right)}}{1909098 \left(n + 45\right)} + \frac{\left(3937060169 n^{2} + 318596027109 n + 6437342987370\right) a{\left(n + 42 \right)}}{28636470 \left(n + 44\right) \left(n + 45\right)} - \frac{20 \left(10642499473 n^{3} + 131130569745 n^{2} + 523101639008 n + 676479670990\right) a{\left(n + 3 \right)}}{954549 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(32308408285 n^{3} + 3921989261040 n^{2} + 158600698992182 n + 2136502268389443\right) a{\left(n + 41 \right)}}{17181882 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{2 \left(1522639518816 n^{3} + 23102179361349 n^{2} + 114623073134811 n + 186316178353624\right) a{\left(n + 4 \right)}}{2863647 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(1609930467320 n^{3} + 190366352222211 n^{2} + 7498484459107459 n + 98389432372490268\right) a{\left(n + 40 \right)}}{171818820 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(3804971677993 n^{3} + 440064029274804 n^{2} + 16963015985846390 n + 217931780731759725\right) a{\left(n + 39 \right)}}{171818820 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(15548159199979 n^{3} + 1835886319671834 n^{2} + 72357012150259889 n + 951786149807346999\right) a{\left(n + 38 \right)}}{515456460 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(26138267987421 n^{3} + 461047351552008 n^{2} + 2671829815477811 n + 5101734103787052\right) a{\left(n + 5 \right)}}{8590941 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(70531330608271 n^{3} + 8602576727959596 n^{2} + 348495164320465598 n + 4690625120329021827\right) a{\left(n + 37 \right)}}{1030912920 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(168349100206419 n^{3} + 17216179047935048 n^{2} + 585719515465402719 n + 6627110172623941078\right) a{\left(n + 35 \right)}}{1374550560 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(207415192065834 n^{3} + 21252035059796023 n^{2} + 729090571325343787 n + 8372155505207484390\right) a{\left(n + 33 \right)}}{687275280 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(278026515625571 n^{3} + 223703296950594003 n^{2} + 10697294752470038176 n + 131893198074222546528\right) a{\left(n + 25 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(360923350174881 n^{3} + 41591954823237812 n^{2} + 1586900114223434109 n + 20067004467949146354\right) a{\left(n + 34 \right)}}{1374550560 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(373964568361927 n^{3} + 43024746558821613 n^{2} + 1648921015477185386 n + 21050688048995889462\right) a{\left(n + 36 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(873937158306953 n^{3} + 15021674793554061 n^{2} + 80516013582548812 n + 131639755119770064\right) a{\left(n + 6 \right)}}{257728230 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(943910568049292 n^{3} + 107340374329873797 n^{2} + 3991984935323713483 n + 48779256844528682352\right) a{\left(n + 32 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(3428586612311169 n^{3} + 314976714458341936 n^{2} + 9618345888443479017 n + 97640481176577770368\right) a{\left(n + 29 \right)}}{687275280 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(4295118684934619 n^{3} + 461525227890974988 n^{2} + 15528709998364411423 n + 167231633396458346700\right) a{\left(n + 27 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(5974283605150013 n^{3} + 584645153185111560 n^{2} + 19062415317643588177 n + 207083043247591955634\right) a{\left(n + 31 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(6021201215316055 n^{3} + 178615602014901720 n^{2} + 1694920131713833823 n + 5169050708308069446\right) a{\left(n + 7 \right)}}{515456460 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(6093757833190537 n^{3} + 420558663151322238 n^{2} + 9229986581619444915 n + 62840128893535771078\right) a{\left(n + 26 \right)}}{1374550560 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(6760721363774609 n^{3} + 662159590698360036 n^{2} + 21542863289783181829 n + 232928600104296438786\right) a{\left(n + 30 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(11948621740392156 n^{3} + 361300051501550677 n^{2} + 3586264383929543011 n + 11669790100614992558\right) a{\left(n + 8 \right)}}{171818820 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(18125723629183009 n^{3} + 1595777321758230174 n^{2} + 46978921963385695943 n + 462189868186756605066\right) a{\left(n + 28 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(33857120742233051 n^{3} + 2488182958991448558 n^{2} + 58837058289881621893 n + 450073268956878488142\right) a{\left(n + 21 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(41358548956713897 n^{3} + 2414397493895746170 n^{2} + 41486348444037248227 n + 220557287301951180602\right) a{\left(n + 12 \right)}}{343637640 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(60374228074610081 n^{3} + 4164180654790905612 n^{2} + 94907623710123121483 n + 714052952648849085960\right) a{\left(n + 24 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(91069057113010309 n^{3} + 6620572732420114047 n^{2} + 160497663450695349626 n + 1297912916247190904586\right) a{\left(n + 23 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(110773412669434655 n^{3} + 4613795088178764100 n^{2} + 62684519790272937867 n + 278047220275893792046\right) a{\left(n + 11 \right)}}{343637640 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(142611573036209141 n^{3} + 10547582020353099960 n^{2} + 259446162104944006885 n + 2122615680268763464950\right) a{\left(n + 22 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(160386506756154695 n^{3} + 5877349535975013144 n^{2} + 70894532725096156501 n + 281132862476082708642\right) a{\left(n + 10 \right)}}{515456460 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(191361611451365597 n^{3} + 6328349532075225234 n^{2} + 68948246823519986803 n + 247106788034858678454\right) a{\left(n + 9 \right)}}{1030912920 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(239702440271620103 n^{3} + 7793125433023212993 n^{2} + 78936895663399910362 n + 240032508355991590608\right) a{\left(n + 13 \right)}}{1030912920 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(317408702570920247 n^{3} + 15042103191334745322 n^{2} + 233616128960390754673 n + 1188030502564368414330\right) a{\left(n + 17 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(332488581364272001 n^{3} + 20871392589302127630 n^{2} + 440860851577259845199 n + 3134883032303929681806\right) a{\left(n + 20 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(928771434194347057 n^{3} + 55466384697979278618 n^{2} + 1109680647392815744811 n + 7441354949041732498206\right) a{\left(n + 19 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(1067515714676291510 n^{3} + 45529693466075607375 n^{2} + 650343555443829200659 n + 3119524363994787821034\right) a{\left(n + 14 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(1113945795335933335 n^{3} + 62851649212712927610 n^{2} + 1186972227629414700593 n + 7508920546982708893770\right) a{\left(n + 18 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(1126898335020999229 n^{3} + 60508445391104373750 n^{2} + 1086644804424638722451 n + 6518683745259726450414\right) a{\left(n + 16 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(2207899984268875345 n^{3} + 106365972500518359468 n^{2} + 1718808522526791287237 n + 9316910813805157288410\right) a{\left(n + 15 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)}, \quad n \geq 44\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 110\)
\(\displaystyle a(6) = 533\)
\(\displaystyle a(7) = 2658\)
\(\displaystyle a(8) = 13527\)
\(\displaystyle a(9) = 69934\)
\(\displaystyle a(10) = 366181\)
\(\displaystyle a(11) = 1937809\)
\(\displaystyle a(12) = 10348422\)
\(\displaystyle a(13) = 55702889\)
\(\displaystyle a(14) = 301933269\)
\(\displaystyle a(15) = 1646763624\)
\(\displaystyle a(16) = 9031253272\)
\(\displaystyle a(17) = 49775122555\)
\(\displaystyle a(18) = 275555847961\)
\(\displaystyle a(19) = 1531630492953\)
\(\displaystyle a(20) = 8544442259905\)
\(\displaystyle a(21) = 47825277966469\)
\(\displaystyle a(22) = 268504431166819\)
\(\displaystyle a(23) = 1511668242788459\)
\(\displaystyle a(24) = 8532497243252199\)
\(\displaystyle a(25) = 48275286710253950\)
\(\displaystyle a(26) = 273732484745996878\)
\(\displaystyle a(27) = 1555293598061318839\)
\(\displaystyle a(28) = 8853638333167138919\)
\(\displaystyle a(29) = 50489303711252247215\)
\(\displaystyle a(30) = 288400078142183511672\)
\(\displaystyle a(31) = 1649926475978732728306\)
\(\displaystyle a(32) = 9452919852620993040184\)
\(\displaystyle a(33) = 54232808019471321040944\)
\(\displaystyle a(34) = 311543517028359215062863\)
\(\displaystyle a(35) = 1791861729398854125336248\)
\(\displaystyle a(36) = 10317883180154667918146993\)
\(\displaystyle a(37) = 59477201630628447417042871\)
\(\displaystyle a(38) = 343209875284884106051289402\)
\(\displaystyle a(39) = 1982420205876416424049736851\)
\(\displaystyle a(40) = 11461392237015753534519534229\)
\(\displaystyle a(41) = 66323171892577862281148462095\)
\(\displaystyle a(42) = 384115075132653459782420361776\)
\(\displaystyle a(43) = 2226428261782638194179854356968\)
\(\displaystyle a{\left(n + 44 \right)} = - \frac{7170000 n \left(n + 1\right) \left(n + 2\right) a{\left(n \right)}}{106061 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{3000 \left(n + 1\right) \left(n + 2\right) \left(75444 n + 243221\right) a{\left(n + 1 \right)}}{106061 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{100 \left(n + 2\right) \left(92067333 n^{2} + 674308332 n + 1191810139\right) a{\left(n + 2 \right)}}{318183 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(11568193 n + 511623053\right) a{\left(n + 43 \right)}}{1909098 \left(n + 45\right)} + \frac{\left(3937060169 n^{2} + 318596027109 n + 6437342987370\right) a{\left(n + 42 \right)}}{28636470 \left(n + 44\right) \left(n + 45\right)} - \frac{20 \left(10642499473 n^{3} + 131130569745 n^{2} + 523101639008 n + 676479670990\right) a{\left(n + 3 \right)}}{954549 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(32308408285 n^{3} + 3921989261040 n^{2} + 158600698992182 n + 2136502268389443\right) a{\left(n + 41 \right)}}{17181882 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{2 \left(1522639518816 n^{3} + 23102179361349 n^{2} + 114623073134811 n + 186316178353624\right) a{\left(n + 4 \right)}}{2863647 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(1609930467320 n^{3} + 190366352222211 n^{2} + 7498484459107459 n + 98389432372490268\right) a{\left(n + 40 \right)}}{171818820 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(3804971677993 n^{3} + 440064029274804 n^{2} + 16963015985846390 n + 217931780731759725\right) a{\left(n + 39 \right)}}{171818820 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(15548159199979 n^{3} + 1835886319671834 n^{2} + 72357012150259889 n + 951786149807346999\right) a{\left(n + 38 \right)}}{515456460 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(26138267987421 n^{3} + 461047351552008 n^{2} + 2671829815477811 n + 5101734103787052\right) a{\left(n + 5 \right)}}{8590941 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(70531330608271 n^{3} + 8602576727959596 n^{2} + 348495164320465598 n + 4690625120329021827\right) a{\left(n + 37 \right)}}{1030912920 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(168349100206419 n^{3} + 17216179047935048 n^{2} + 585719515465402719 n + 6627110172623941078\right) a{\left(n + 35 \right)}}{1374550560 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(207415192065834 n^{3} + 21252035059796023 n^{2} + 729090571325343787 n + 8372155505207484390\right) a{\left(n + 33 \right)}}{687275280 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(278026515625571 n^{3} + 223703296950594003 n^{2} + 10697294752470038176 n + 131893198074222546528\right) a{\left(n + 25 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(360923350174881 n^{3} + 41591954823237812 n^{2} + 1586900114223434109 n + 20067004467949146354\right) a{\left(n + 34 \right)}}{1374550560 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(373964568361927 n^{3} + 43024746558821613 n^{2} + 1648921015477185386 n + 21050688048995889462\right) a{\left(n + 36 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(873937158306953 n^{3} + 15021674793554061 n^{2} + 80516013582548812 n + 131639755119770064\right) a{\left(n + 6 \right)}}{257728230 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(943910568049292 n^{3} + 107340374329873797 n^{2} + 3991984935323713483 n + 48779256844528682352\right) a{\left(n + 32 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(3428586612311169 n^{3} + 314976714458341936 n^{2} + 9618345888443479017 n + 97640481176577770368\right) a{\left(n + 29 \right)}}{687275280 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(4295118684934619 n^{3} + 461525227890974988 n^{2} + 15528709998364411423 n + 167231633396458346700\right) a{\left(n + 27 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(5974283605150013 n^{3} + 584645153185111560 n^{2} + 19062415317643588177 n + 207083043247591955634\right) a{\left(n + 31 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(6021201215316055 n^{3} + 178615602014901720 n^{2} + 1694920131713833823 n + 5169050708308069446\right) a{\left(n + 7 \right)}}{515456460 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(6093757833190537 n^{3} + 420558663151322238 n^{2} + 9229986581619444915 n + 62840128893535771078\right) a{\left(n + 26 \right)}}{1374550560 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(6760721363774609 n^{3} + 662159590698360036 n^{2} + 21542863289783181829 n + 232928600104296438786\right) a{\left(n + 30 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(11948621740392156 n^{3} + 361300051501550677 n^{2} + 3586264383929543011 n + 11669790100614992558\right) a{\left(n + 8 \right)}}{171818820 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(18125723629183009 n^{3} + 1595777321758230174 n^{2} + 46978921963385695943 n + 462189868186756605066\right) a{\left(n + 28 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(33857120742233051 n^{3} + 2488182958991448558 n^{2} + 58837058289881621893 n + 450073268956878488142\right) a{\left(n + 21 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(41358548956713897 n^{3} + 2414397493895746170 n^{2} + 41486348444037248227 n + 220557287301951180602\right) a{\left(n + 12 \right)}}{343637640 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(60374228074610081 n^{3} + 4164180654790905612 n^{2} + 94907623710123121483 n + 714052952648849085960\right) a{\left(n + 24 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(91069057113010309 n^{3} + 6620572732420114047 n^{2} + 160497663450695349626 n + 1297912916247190904586\right) a{\left(n + 23 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(110773412669434655 n^{3} + 4613795088178764100 n^{2} + 62684519790272937867 n + 278047220275893792046\right) a{\left(n + 11 \right)}}{343637640 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(142611573036209141 n^{3} + 10547582020353099960 n^{2} + 259446162104944006885 n + 2122615680268763464950\right) a{\left(n + 22 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(160386506756154695 n^{3} + 5877349535975013144 n^{2} + 70894532725096156501 n + 281132862476082708642\right) a{\left(n + 10 \right)}}{515456460 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(191361611451365597 n^{3} + 6328349532075225234 n^{2} + 68948246823519986803 n + 247106788034858678454\right) a{\left(n + 9 \right)}}{1030912920 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(239702440271620103 n^{3} + 7793125433023212993 n^{2} + 78936895663399910362 n + 240032508355991590608\right) a{\left(n + 13 \right)}}{1030912920 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(317408702570920247 n^{3} + 15042103191334745322 n^{2} + 233616128960390754673 n + 1188030502564368414330\right) a{\left(n + 17 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(332488581364272001 n^{3} + 20871392589302127630 n^{2} + 440860851577259845199 n + 3134883032303929681806\right) a{\left(n + 20 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(928771434194347057 n^{3} + 55466384697979278618 n^{2} + 1109680647392815744811 n + 7441354949041732498206\right) a{\left(n + 19 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(1067515714676291510 n^{3} + 45529693466075607375 n^{2} + 650343555443829200659 n + 3119524363994787821034\right) a{\left(n + 14 \right)}}{2061825840 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} + \frac{\left(1113945795335933335 n^{3} + 62851649212712927610 n^{2} + 1186972227629414700593 n + 7508920546982708893770\right) a{\left(n + 18 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(1126898335020999229 n^{3} + 60508445391104373750 n^{2} + 1086644804424638722451 n + 6518683745259726450414\right) a{\left(n + 16 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)} - \frac{\left(2207899984268875345 n^{3} + 106365972500518359468 n^{2} + 1718808522526791287237 n + 9316910813805157288410\right) a{\left(n + 15 \right)}}{4123651680 \left(n + 43\right) \left(n + 44\right) \left(n + 45\right)}, \quad n \geq 44\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 140 rules.
Finding the specification took 7708 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{135}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{7}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{21}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= y x\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{26}\! \left(x , y\right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= y F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{35}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\
F_{57}\! \left(x \right) &= 0\\
F_{58}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{59}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{57}\! \left(x \right)+F_{62}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{35}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{64}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= 2 F_{57}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{56}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{69}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= y F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{56}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{75}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)+F_{57}\! \left(x \right)+F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= 2 F_{57}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{7}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{91}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= -\frac{-F_{96}\! \left(x , y\right)+F_{96}\! \left(x , 1\right)}{-1+y}\\
F_{96}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{98}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{2}\! \left(x \right) F_{21}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{24}\! \left(x \right) F_{26}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{24}\! \left(x \right) F_{32}\! \left(x , y\right)\\
F_{101}\! \left(x , y\right) &= y F_{102}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{115}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{123}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{115}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{112}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right) F_{111}\! \left(x \right)\\
F_{110}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{92}\! \left(x , y\right)\\
F_{111}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{92}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{111}\! \left(x \right)+F_{21}\! \left(x , y\right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= F_{115}\! \left(x \right) F_{125}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)+F_{132}\! \left(x , y\right)\\
F_{126}\! \left(x , y\right) &= F_{127}\! \left(x , y\right)+F_{131}\! \left(x , y\right)\\
F_{127}\! \left(x , y\right) &= F_{128}\! \left(x \right)+F_{129}\! \left(x , y\right)\\
F_{128}\! \left(x \right) &= F_{111}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\
F_{130}\! \left(x , y\right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right) F_{92}\! \left(x , y\right)\\
F_{131}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{91}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\
F_{133}\! \left(x , y\right) &= F_{111}\! \left(x \right) F_{90}\! \left(x , y\right)\\
F_{134}\! \left(x , y\right) &= -\frac{-F_{129}\! \left(x , y\right)+F_{129}\! \left(x , 1\right)}{-1+y}\\
F_{135}\! \left(x , y\right) &= F_{136}\! \left(x , y\right)\\
F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{138}\! \left(x , y\right) &= F_{137}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{139}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= -\frac{y \left(F_{7}\! \left(x , 1\right)-F_{7}\! \left(x , y\right)\right)}{-1+y}\\
\end{align*}\)