0231_0321_1302

Counting sequence:
1, 1, 2, 6, 21, 78, 301, 1197, 4875, 20235, 85294, 364131, 1571212, 6841633, 30025137, 132668839, 589726354, 2635305193, 11832046239, 53348943586, 241461778944, 1096661157410, 4996474775550, 22829953459296, 104590869040121, 480329662190992, 2210862354275757, 10197394824448699, 47125733468184148, 218178033026302757, 1011803679167308061, 4699666988943085822, 21861558341620036344, 101835631421021432358, 474993649331180829876, 2218262908990703549485, 10371601868178231726339, 48546628165679553004883, 227471847701161481929580, 1066911600686639761648400, 5008872608402314452808170, 23536529555954712382876080, 110692269422490621222654210, 521011598383858544239282930, 2454238137606725133641362840, 11569384467318156903924281870, 54577552214224800402191665790, 257641009276254385468072298705, 1217028564667881342593932384350, 5752542620910016706677972104515, 27207043579201306150706558877291, 128752321341336274665670860396679, 609637049997520839165607055603863, 2888159078388723917484174932445357, 13689757195076008475101110572694094, 64921288346807244900579304228979352, 308026135538121411810877378430444549, 1462142716002717365496897650922923608, 6943633744341022528281660650412939489, 32989225458380810177164060686492253743, 156797619653716188556605144250351354292, 745560528926813775714872805388461911345, 3546474125176654732088759981692522931705, 16876242061690798110167080909293321402804, 80336818163867156747152010765941570741451, 382567913507833802401602519511516886376192, 1822438230392519499906161404636302438195097, 8684463321829286423383311610485167097348924, 41397563021296180366751877734651080369235547, 197398616110782367031494747557074261451031108, 941557955186655371123481394660701724911068057, 4492415213456050840290846667113915967023973655, 21440702408410410723705806982206695637938702550, 102357806743780144427610719212084072842908133873, 488790179298745470165433606415464276229116329107, 2334749736916837392306601257826109620749210810704, 11155049793981959824425239062412090604335345973884, 53310539184510237164486939072947899210108741080268, 254836834592633265599561233193751540952736921386224, 1218473762806993785541480615038569689724464034658554, 5827367461053301058772089352226623797541410283706192, 27875865173266677565994274178363078696333510155860254, 133377204717434580735123525050267652360114242547096938, 638307338146755837613928921243088535639961890340927030, 3055419038829497854947091629993727200621408998091928930, 14628583258166826500368317459532671256527199865121003346, 70052268175261788339644704349038567570765635106264071264, 335527836154072358475065678779726938772119636953454625328, 1607383013218074038450415491814982012985485506129335375002, 7701809377249368418687929438569876696669382140224957710848, 36910242703125990936257670515567203233410873434120396779546, 176921255429956726230721575467500777782216033616418880932026, 848184612357434198395056803868393532782273249970025912456492, 4067020631948180040531246087863537966943691664443935600863846, 19504569457673743765111889769217864713686282633265645256167874, 93555390175466719725909345318086870246327287359691254137458806, 448820008634303936166331651097289014105619368139178690489180155, 2153501092481852502995217527708053341970870326752339198314328130, 10334418885117196017794702649091509935456033777325079408668005169, 49601376878542975759968787031751850434201839129982391631859534801, 238104048971687759836078698557662670287633454636699040925912572030

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

Implicit equation for the generating function in latex syntax:
x^{2} F \! \left(x \right)^{3}-x \left(2 x -1\right) F \! \left(x \right)^{2}+\left(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) = 21
a(5) = 78
a(6) = 301
a(7) = 1197
a(n+8) = 128/25*n*(2*n+1)/(n+9)/(n+8)*a(n)-8/25*(137*n^2+340*n+192)/(n+9)/(n+8)*a(n+1)+3/25*(629*n^2+2585*n+2496)/(n+9)/(n+8)*a(n+2)-2/25*(808*n^2+4145*n+4515)/(n+9)/(n+8)*a(n+3)+1/25*(437*n^2+877*n-5280)/(n+9)/(n+8)*a(n+4)+2/25*(266*n^2+3781*n+12885)/(n+9)/(n+8)*a(n+5)-1/5*(7*n+51)*(17*n+112)/(n+9)/(n+8)*a(n+6)+4/5*(11*n+81)/(n+9)*a(n+7), n >= 8

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) = 21
a \! \left(5\right) = 78
a \! \left(6\right) = 301
a \! \left(7\right) = 1197
a \! \left(n +8\right) = \frac{128 n \left(2 n +1\right) a \! \left(n \right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{8 \left(137 n^{2}+340 n +192\right) a \! \left(n +1\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{3 \left(629 n^{2}+2585 n +2496\right) a \! \left(n +2\right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{2 \left(808 n^{2}+4145 n +4515\right) a \! \left(n +3\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{\left(437 n^{2}+877 n -5280\right) a \! \left(n +4\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{2 \left(266 n^{2}+3781 n +12885\right) a \! \left(n +5\right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{\left(7 n +51\right) \left(17 n +112\right) a \! \left(n +6\right)}{5 \left(n +9\right) \left(n +8\right)}+\frac{4 \left(11 n +81\right) a \! \left(n +7\right)}{5 \left(n +9\right)}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/347/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[8,x]+F[9,x]
F[8,x] = F[0,x]*F[4,x]
F[9,x] = F[10,x]
F[10,x] = F[0,x]^2*F[11,x]*F[2,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[11,x]*F[13,x]
F[13,x] = 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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)+F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \left(x \right) F_{4}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{0} \left(x \right)^{2} F_{11}\! \left(x \right) F_{2}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
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_13(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_13(x)*F_7(x))
Eq(F_7(x), F_8(x) + F_9(x))
Eq(F_8(x), F_0(x)*F_4(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_0(x)**2*F_11(x)*F_2(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_11(x)*F_13(x))
Eq(F_13(x), 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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 2], [0, 2], [0, 0], [0, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 1]]}], "ignore_parent": false, "strategy_class": 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