02341_02413_02431_03241_03412_03421_04123_04132_04213_04231_04312_04321_20341_20413_20431

Counting sequence:
1, 1, 2, 6, 24, 105, 479, 2264, 11014, 54781, 277243, 1423072, 7390868, 38767770, 205082371, 1092873903, 5861285512, 31613001003, 171361532780, 933048862378, 5100850589029, 27987236928541, 154067774624156, 850691657199344, 4710111484437920, 26145214890038288, 145468802317447027, 811125938578955092, 4531894994957530532, 25367863933131308811, 142247932477975240060, 798944896130824784761, 4494195187383295622794, 25316915229688616806247, 142809304566974629743208, 806594705354396460189988, 4561172378069594267875982, 25822095821536394154877081, 146343775183946813629823303, 830234318094439310414483598, 4714639425346937724855522139, 26797668720964168220214129199, 152449886800703384294598362047, 868002422625447673716830136908, 4946088746880138760343908975040, 28205546366226860541216421595180, 160962347494041257599775102759736, 919216257395514384355030980244452, 5252934873499071177867843003748472, 30037607471141381685180312752622840, 171868568525232156838083868331257844, 983976764717815755123152452131075808, 5636643382064660228609617365568761664, 32306822173956348540289478328875312354, 185266564046753213257826550005917651623, 1062969044945487638233707100287175788435, 6101786984706146731814680634905648978558, 35042802345604186953424673470414466229111, 201344084501864455602099135425465585262061, 1157365247090915160684239505747467160292022, 6655598489996682578334670977649978292712539, 38289775384478779967904943284514067807137046, 220369637885261816751049770549059151544214266, 1268786114971953747489473212880814399163978201, 7307814932242216919350198485788103325490614070, 42106012484252436018098732191043453058397940295, 242690824049498710343411629157122441050135477016, 1399299552616355267449987497560473885557741928758, 8070710411877364100352093674351753392397794174500, 46564225855805264715677087949433713125665468821405, 268737681197582203884117285597455251175014858032889, 1551445313250184641760921957506243235193274012749334, 8959266137537411698907758811052695196915356391919876, 51752692086621661263261733749435673936631567470638977, 299029904882423855298355351528678787116024493242102317, 1728280496219759703049013168350974152471249134804806792, 9991453096416323924458385435633363521539564999138562336, 57776989456022879107316612728292714414536924022971507080, 334187436477962356067616319500930425174558707970698851489, 1933443654688442685299972686916931009309645832865930209433, 11188618136933959320817568722848976924060545613239687973509, 64762313082803561599289498265168101359678256837152549849361, 374944297025488294133079343307334629884138027056056031975706, 2171237216870517691452647435134534514664453045924226743758334, 12575973837278582265228679218578172073783269647779994320023345, 72856380524177023453022541755693405709435566355724484912426317, 422165791934555184896663947139540573883685710570584209052078636, 2446729211258490743322527337527194095688210253992806654979871800, 14183199229772331496338440147255844704166927309032266673923434904, 82232975549131984078425467127523124326423418741932690657538011892, 476869397545389839676275612558194945868601612278162588081156242424, 2765876274826694320305894123004879955129724095045627930831540647152, 16045163760567841803742667492851870319998484294830646910879644472808, 93096206552731270237477787961004700360416278877571898438768151361604, 540249797346840470987380286533547204418025136080205588615886095713708, 3135670784631074367390731931513098628227525321511668345472190940659496, 18202791629599607853801082975525208590288518886392914462278567385963720, 105685582608374044583910752232541655193302563913291020200109934859702944, 613708684219566000453446932789633712064166336286611317418245907340796952, 3564315789776644722585759006701442020084098519259867917654591426722187016, 20704088389029382053686617864651898566427397388363802531210633087527112300

Implicit equation for the generating function in Maple syntax:
3*x^2*F(x)^4-x*(7*x-2)*F(x)^3+(5*x-4)*x*F(x)^2+(-x^2+4*x-1)*F(x)-x+1 = 0

Implicit equation for the generating function in latex syntax:
3 x^{2} F \! \left(x \right)^{4}-x \left(7 x -2\right) F \! \left(x \right)^{3}+\left(5 x -4\right) x F \! \left(x \right)^{2}+\left(-x^{2}+4 x -1\right) F \! \left(x \right)-x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 105
a(6) = 479
a(7) = 2264
a(8) = 11014
a(9) = 54781
a(10) = 277243
a(11) = 1423072
a(12) = 7390868
a(13) = 38767770
a(14) = 205082371
a(15) = 1092873903
a(16) = 5861285512
a(17) = 31613001003
a(18) = 171361532780
a(19) = 933048862378
a(n+20) = 17/14*n*(2*n+1)*(n+1)/(n+20)/(2*n+41)/(n+21)*a(n)+51/224*(n+1)*(89*n^2+38*n-136)/(n+20)/(2*n+41)/(n+21)*a(n+1)-3/896*(55013*n^3+579531*n^2+1725298*n+1566048)/(n+20)/(2*n+41)/(n+21)*a(n+2)-1/448*(1154927*n^3+12489522*n^2+44008339*n+50705526)/(n+20)/(2*n+41)/(n+21)*a(n+3)-1/112*(1350586*n^3+18085830*n^2+80401646*n+118607025)/(n+20)/(2*n+41)/(n+21)*a(n+4)-1/896*(30203209*n^3+527621856*n^2+3034053839*n+5752317216)/(n+20)/(2*n+41)/(n+21)*a(n+5)-1/224*(12385916*n^3+254881968*n^2+1715994646*n+3793002999)/(n+20)/(2*n+41)/(n+21)*a(n+6)-1/448*(1730279*n^3-46213812*n^2-985848521*n-4048292886)/(n+20)/(2*n+41)/(n+21)*a(n+7)+1/896*(209904451*n^3+5284434081*n^2+44335035926*n+123972535656)/(n+20)/(2*n+41)/(n+21)*a(n+8)-1/896*(145843697*n^3+3837620745*n^2+33772023016*n+99506439720)/(n+20)/(2*n+41)/(n+21)*a(n+9)-1/448*(40689028*n^3+1544738127*n^2+18614492399*n+72251381310)/(n+20)/(2*n+41)/(n+21)*a(n+10)+1/896*(200384015*n^3+7100323410*n^2+83100816883*n+321382863588)/(n+20)/(2*n+41)/(n+21)*a(n+11)-1/896*(162922858*n^3+5908690971*n^2+70942611521*n+281756533740)/(n+20)/(2*n+41)/(n+21)*a(n+12)+1/448*(36070427*n^3+1364469792*n^2+17059995535*n+70385229138)/(n+20)/(2*n+41)/(n+21)*a(n+13)-1/224*(4195513*n^3+168766410*n^2+2250519242*n+9940719990)/(n+20)/(2*n+41)/(n+21)*a(n+14)+1/896*(2583997*n^3+124574688*n^2+2021589587*n+11028864648)/(n+20)/(2*n+41)/(n+21)*a(n+15)-1/224*(458995*n^3+24644715*n^2+440393444*n+2618859564)/(n+20)/(2*n+41)/(n+21)*a(n+16)+1/448*(548447*n^3+29959344*n^2+545382121*n+3308471928)/(n+20)/(2*n+41)/(n+21)*a(n+17)-3/224*(24613*n^3+1393108*n^2+26282463*n+165274488)/(n+20)/(2*n+41)/(n+21)*a(n+18)+1/112*(4637*n^2+180800*n+1762323)/(n+21)/(2*n+41)*a(n+19), n >= 20

Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 24
a \! \left(5\right) = 105
a \! \left(6\right) = 479
a \! \left(7\right) = 2264
a \! \left(8\right) = 11014
a \! \left(9\right) = 54781
a \! \left(10\right) = 277243
a \! \left(11\right) = 1423072
a \! \left(12\right) = 7390868
a \! \left(13\right) = 38767770
a \! \left(14\right) = 205082371
a \! \left(15\right) = 1092873903
a \! \left(16\right) = 5861285512
a \! \left(17\right) = 31613001003
a \! \left(18\right) = 171361532780
a \! \left(19\right) = 933048862378
a \! \left(n +20\right) = \frac{17 n \left(2 n +1\right) \left(n +1\right) a \! \left(n \right)}{14 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{51 \left(n +1\right) \left(89 n^{2}+38 n -136\right) a \! \left(n +1\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{3 \left(55013 n^{3}+579531 n^{2}+1725298 n +1566048\right) a \! \left(n +2\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1154927 n^{3}+12489522 n^{2}+44008339 n +50705526\right) a \! \left(n +3\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1350586 n^{3}+18085830 n^{2}+80401646 n +118607025\right) a \! \left(n +4\right)}{112 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(30203209 n^{3}+527621856 n^{2}+3034053839 n +5752317216\right) a \! \left(n +5\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(12385916 n^{3}+254881968 n^{2}+1715994646 n +3793002999\right) a \! \left(n +6\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(1730279 n^{3}-46213812 n^{2}-985848521 n -4048292886\right) a \! \left(n +7\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(209904451 n^{3}+5284434081 n^{2}+44335035926 n +123972535656\right) a \! \left(n +8\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(145843697 n^{3}+3837620745 n^{2}+33772023016 n +99506439720\right) a \! \left(n +9\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(40689028 n^{3}+1544738127 n^{2}+18614492399 n +72251381310\right) a \! \left(n +10\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(200384015 n^{3}+7100323410 n^{2}+83100816883 n +321382863588\right) a \! \left(n +11\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(162922858 n^{3}+5908690971 n^{2}+70942611521 n +281756533740\right) a \! \left(n +12\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(36070427 n^{3}+1364469792 n^{2}+17059995535 n +70385229138\right) a \! \left(n +13\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(4195513 n^{3}+168766410 n^{2}+2250519242 n +9940719990\right) a \! \left(n +14\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(2583997 n^{3}+124574688 n^{2}+2021589587 n +11028864648\right) a \! \left(n +15\right)}{896 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{\left(458995 n^{3}+24644715 n^{2}+440393444 n +2618859564\right) a \! \left(n +16\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(548447 n^{3}+29959344 n^{2}+545382121 n +3308471928\right) a \! \left(n +17\right)}{448 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}-\frac{3 \left(24613 n^{3}+1393108 n^{2}+26282463 n +165274488\right) a \! \left(n +18\right)}{224 \left(n +20\right) \left(2 n +41\right) \left(n +21\right)}+\frac{\left(4637 n^{2}+180800 n +1762323\right) a \! \left(n +19\right)}{112 \left(n +21\right) \left(2 n +41\right)}, \quad n \geq 20

Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/24314/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[11,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[11,x]*F[7,x]
F[7,x] = F[12,x]+F[8,x]
F[8,x] = F[0,x]*F[9,x]
F[9,x] = 1/F[11,x]*F[10,x]
F[10,x] = F[2,x]
F[11,x] = x
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]*F[4,x]
F[14,x] = 1/F[11,x]*F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[11,x]*F[17,x]*F[9,x]
F[17,x] = 1/F[11,x]/F[4,x]*F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]+F[27,x]
F[20,x] = F[21,x]+F[25,x]
F[21,x] = F[22,x]
F[22,x] = F[11,x]*F[23,x]*F[4,x]
F[23,x] = F[0,x]+F[24,x]
F[24,x] = F[2,x]
F[25,x] = F[26,x]
F[26,x] = F[0,x]^2*F[11,x]*F[2,x]
F[27,x] = F[28,x]
F[28,x] = F[11,x]*F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[0,x]*F[11,x]*F[23,x]*F[4,x]
System of equations in latex syntax:
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_{4}\! \left(x \right)
F_{14}\! \left(x \right) = \frac{F_{15}\! \left(x \right)}{F_{11}\! \left(x \right)}
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = \frac{F_{18}\! \left(x \right)}{F_{11}\! \left(x \right) F_{4}\! \left(x \right)}
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \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_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{0}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{0} \left(x \right)^{2} F_{11}\! \left(x \right) F_{2}\! \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) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_11(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_11(x)*F_7(x))
Eq(F_7(x), F_12(x) + F_8(x))
Eq(F_8(x), F_0(x)*F_9(x))
Eq(F_9(x), F_10(x)/F_11(x))
Eq(F_10(x), F_2(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x)*F_4(x))
Eq(F_14(x), F_15(x)/F_11(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_11(x)*F_17(x)*F_9(x))
Eq(F_17(x), F_18(x)/(F_11(x)*F_4(x)))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x) + F_27(x))
Eq(F_20(x), F_21(x) + F_25(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_11(x)*F_23(x)*F_4(x))
Eq(F_23(x), F_0(x) + F_24(x))
Eq(F_24(x), F_2(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_0(x)**2*F_11(x)*F_2(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_11(x)*F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_0(x)*F_11(x)*F_23(x)*F_4(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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