0132_0231_1023_1230_2013_3012

Counting sequence:
1, 1, 2, 6, 18, 49, 143, 445, 1452, 4891, 16833, 58832, 208068, 742967, 2674519, 9694937, 35357776, 129644911, 477638837, 1767263344, 6564120592, 24466267211, 91482563851, 343059613882, 1289904147578, 4861946401729, 18367353072453, 69533550916330, 263747951750712, 1002242216651747, 3814986502092711, 14544636039227345, 55534064877048664, 212336130412243607, 812944042149731293, 3116285494907301824, 11959798385860454088, 45950804324621742995, 176733862787006702067, 680425371729975801094, 2622127042276492109562, 10113918591637898134801, 39044429911904443960061, 150853479205085351661562, 583300119592996693088944, 2257117854077248073254667, 8740328711533173390047311, 33868773757191046886430526, 131327898242169365477992982, 509552245179617138054609701, 1978261657756160653623775633, 7684785670514316385230817382, 29869166945772625950142418788, 116157871455782434250553847207, 451959718027953471447609510803, 1759414616608818870992479877404, 6852456927844873497549658465798, 26700952856774851904245220914205, 104088460289122304033498318813677, 405944995127576985730643443368766, 1583850964596120042686772779040608, 6182127958584855650487080847218107, 24139737743045626825711458546275143, 94295850558771979787935384946382017, 368479169875816659479009042713548904, 1440418573150919668872489894243867367, 5632681584560312734993915705849147181, 22033725021956517463358552614056952096, 86218923998960285726185640663701110712, 337485502510215975556783793455058626979, 1321422108420282270489942177190229546947, 5175569924646105559418940193995065718766, 20276890389709399862928998568254641028186, 79463489365077377841208237632349268887057, 311496878311103321137536291518809134029869, 1221395654430378811828760722007962130793722, 4790408930363303911328386208394864461027296, 18793142726809884575211361279087545193252891, 73745243611532458459690151854647329239338527, 289450081175264899454283846029490767264395234, 1136359577947336271931632877004667456667617022, 4462290049988320482463241297506133183499657901, 17526585015616776834735140517915655636396237521, 68854441132780194707888052034668647142985209422, 270557451039395118028642463289168566420671283844, 1063353702922273835973036658043476458723103408007, 4180080073556524734514695828170907458428751317891, 16435314834665426797069144960762886143367590398596, 64633260585762914370496637486146181462681535264742, 254224158304000796523953440778841647086547372030429, 1000134600800354781929399250536541864362461089954717, 3935312233584004685417853572763349509774031680027806, 15487357822491889407128326963778343232013931127839696, 60960876535340415751462563580829648891969728907442187, 239993345518077005168915776623476723006280827488233879, 944973797977428207852605870454939596837230758234908422, 3721443204405954385563870541379246659709506697378698766, 14657929356129575437016877846657032761712954950899759661, 57743358069601357782187700608042856334020731624756615657, 227508830794229349661819540395688853956041682601541052094, 896519947090131496687170070074100632420837521538745914172

Generating function in Maple syntax:
1/2*(-(x-1)^3*(-4*x+1)^(1/2)-2*x^7+4*x^6-2*x^5-2*x^4+x^3-3*x^2+3*x-1)/x/(x-1)^3

Generating function in latex syntax:
\frac{-\left(x -1\right)^{3} \sqrt{-4 x +1}-2 x^{7}+4 x^{6}-2 x^{5}-2 x^{4}+x^{3}-3 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-x**7 + 2*x**6 - x**5 - x**4 + x**3/2 - 3*x**2/2 + 3*x/2 - sqrt(1 - 4*x)*(x - 1)**3/2 - 1/2)/(x*(x - 1)**3)

Implicit equation for the generating function in Maple syntax:
x*(x-1)^6*F(x)^2+(2*x^7-4*x^6+2*x^5+2*x^4-x^3+3*x^2-3*x+1)*(x-1)^3*F(x)+x^13-4*x^12+6*x^11-2*x^10-4*x^9+7*x^8-9*x^7+10*x^6-8*x^5+13*x^4-19*x^3+15*x^2-6*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{6} F \! \left(x \right)^{2}+\left(2 x^{7}-4 x^{6}+2 x^{5}+2 x^{4}-x^{3}+3 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{13}-4 x^{12}+6 x^{11}-2 x^{10}-4 x^{9}+7 x^{8}-9 x^{7}+10 x^{6}-8 x^{5}+13 x^{4}-19 x^{3}+15 x^{2}-6 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 49
a(6) = 143
a(7) = 445
a(8) = 1452
a(n+1) = 2*(2*n+1)/(n+2)*a(n)-1/2*(3*n^3-11*n^2+10*n+4)/(n+2), n >= 9

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) = 18
a \! \left(5\right) = 49
a \! \left(6\right) = 143
a \! \left(7\right) = 445
a \! \left(8\right) = 1452
a \! \left(n +1\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{3 n^{3}-11 n^{2}+10 n +4}{2 \left(n +2\right)}, \quad n \geq 9

Specification 1
Strategy pack name: requirement_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22358/
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[4,x]*F[9,x]
F[4,x] = F[46,x]+F[5,x]
F[5,x] = F[10,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[6,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]+F[30,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[9,x]
F[13,x] = F[14,x]+F[22,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[9,x]
F[21,x] = F[7,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[9,x]
F[25,x] = F[21,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[9,x]
F[29,x] = F[27,x]+F[7,x]
F[30,x] = F[31,x]+F[32,x]+F[40,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[9,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[35,x]+F[7,x]
F[35,x] = F[28,x]
F[36,x] = F[37,x]+F[42,x]
F[37,x] = F[31,x]+F[38,x]+F[40,x]
F[38,x] = F[39,x]*F[9,x]
F[39,x] = F[37,x]+F[7,x]
F[40,x] = F[41,x]*F[9,x]
F[41,x] = F[37,x]+F[7,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]*F[9,x]
F[44,x] = F[27,x]+F[45,x]
F[45,x] = F[43,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x,1]
F[48,x,k[0]] = F[49,x,k[0]]
F[49,x,k[0]] = F[50,x,k[0]]
F[50,x,k[0]] = F[51,x,k[0]]*F[60,x,k[0]]*F[9,x]
F[51,x,k[0]] = F[52,x,k[0]]+F[70,x,k[0]]
F[52,x,k[0]] = F[41,x]+F[53,x,k[0]]
F[53,x,k[0]] = F[54,x,k[0]]+F[61,x,k[0]]
F[54,x,k[0]] = F[55,x,k[0]]
F[55,x,k[0]] = F[56,x,k[0]]*F[9,x]
F[56,x,k[0]] = F[54,x,k[0]]+F[57,x,k[0]]
F[57,x,k[0]] = F[58,x,k[0]]
F[58,x,k[0]] = F[59,x,k[0]]*F[60,x,k[0]]
F[59,x,k[0]] = k[0]*x
F[60,x,k[0]] = F[1,x]+F[57,x,k[0]]
F[61,x,k[0]] = 2*F[31,x]+F[62,x,k[0]]+F[69,x,k[0]]
F[62,x,k[0]] = F[63,x,k[0]]*F[9,x]
F[63,x,k[0]] = F[61,x,k[0]]+F[64,x,k[0]]
F[64,x,k[0]] = F[31,x]+F[65,x,k[0]]+F[67,x,k[0]]
F[65,x,k[0]] = F[59,x,k[0]]*F[66,x,k[0]]
F[66,x,k[0]] = F[64,x,k[0]]+F[7,x]
F[67,x,k[0]] = F[68,x,k[0]]*F[9,x]
F[68,x,k[0]] = F[57,x,k[0]]+F[64,x,k[0]]
F[69,x,k[0]] = F[53,x,k[0]]*F[9,x]
F[70,x,k[0]] = (F[49,x,k[0]]*k[0]-F[49,x,1])/(-1+k[0])
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_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{46}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{22}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{9}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{9}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right)+F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{40}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{9}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{9}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{9}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)+F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right) F_{9}\! \left(x \right)
F_{44}\! \left(x \right) = F_{27}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x , 1\right)
F_{48}\! \left(x , y\right) = 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_{51}\! \left(x , y\right) F_{60}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{51}\! \left(x , y\right) = F_{52}\! \left(x , y\right)+F_{70}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{41}\! \left(x \right)+F_{53}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{54}\! \left(x , y\right)+F_{61}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{55}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{56}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{56}\! \left(x , y\right) = F_{54}\! \left(x , y\right)+F_{57}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{58}\! \left(x , y\right)
F_{58}\! \left(x , y\right) = F_{59}\! \left(x , y\right) F_{60}\! \left(x , y\right)
F_{59}\! \left(x , y\right) = y x
F_{60}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{57}\! \left(x , y\right)
F_{61}\! \left(x , y\right) = 2 F_{31}\! \left(x \right)+F_{62}\! \left(x , y\right)+F_{69}\! \left(x , y\right)
F_{62}\! \left(x , y\right) = F_{63}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{63}\! \left(x , y\right) = F_{61}\! \left(x , y\right)+F_{64}\! \left(x , y\right)
F_{64}\! \left(x , y\right) = F_{31}\! \left(x \right)+F_{65}\! \left(x , y\right)+F_{67}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = F_{59}\! \left(x , y\right) F_{66}\! \left(x , y\right)
F_{66}\! \left(x , y\right) = F_{64}\! \left(x , y\right)+F_{7}\! \left(x \right)
F_{67}\! \left(x , y\right) = F_{68}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{68}\! \left(x , y\right) = F_{57}\! \left(x , y\right)+F_{64}\! \left(x , y\right)
F_{69}\! \left(x , y\right) = F_{53}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{70}\! \left(x , y\right) = \frac{y F_{49}\! \left(x , y\right)-F_{49}\! \left(x , 1\right)}{-1+y}
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_4(x)*F_9(x))
Eq(F_4(x), F_46(x) + F_5(x))
Eq(F_5(x), F_10(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_6(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x) + F_30(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_9(x))
Eq(F_13(x), F_14(x) + F_22(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_9(x))
Eq(F_21(x), F_7(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_9(x))
Eq(F_25(x), F_21(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_9(x))
Eq(F_29(x), F_27(x) + F_7(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_40(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_9(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_35(x) + F_7(x))
Eq(F_35(x), F_28(x))
Eq(F_36(x), F_37(x) + F_42(x))
Eq(F_37(x), F_31(x) + F_38(x) + F_40(x))
Eq(F_38(x), F_39(x)*F_9(x))
Eq(F_39(x), F_37(x) + F_7(x))
Eq(F_40(x), F_41(x)*F_9(x))
Eq(F_41(x), F_37(x) + F_7(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x)*F_9(x))
Eq(F_44(x), F_27(x) + F_45(x))
Eq(F_45(x), F_43(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x, 1))
Eq(F_48(x, k_0), F_49(x, k_0))
Eq(F_49(x, k_0), F_50(x, k_0))
Eq(F_50(x, k_0), F_51(x, k_0)*F_60(x, k_0)*F_9(x))
Eq(F_51(x, k_0), F_52(x, k_0) + F_70(x, k_0))
Eq(F_52(x, k_0), F_41(x) + F_53(x, k_0))
Eq(F_53(x, k_0), F_54(x, k_0) + F_61(x, k_0))
Eq(F_54(x, k_0), F_55(x, k_0))
Eq(F_55(x, k_0), F_56(x, k_0)*F_9(x))
Eq(F_56(x, k_0), F_54(x, k_0) + F_57(x, k_0))
Eq(F_57(x, k_0), F_58(x, k_0))
Eq(F_58(x, k_0), F_59(x, k_0)*F_60(x, k_0))
Eq(F_59(x, k_0), k_0*x)
Eq(F_60(x, k_0), F_1(x) + F_57(x, k_0))
Eq(F_61(x, k_0), 2*F_31(x) + F_62(x, k_0) + F_69(x, k_0))
Eq(F_62(x, k_0), F_63(x, k_0)*F_9(x))
Eq(F_63(x, k_0), F_61(x, k_0) + F_64(x, k_0))
Eq(F_64(x, k_0), F_31(x) + F_65(x, k_0) + F_67(x, k_0))
Eq(F_65(x, k_0), F_59(x, k_0)*F_66(x, k_0))
Eq(F_66(x, k_0), F_64(x, k_0) + F_7(x))
Eq(F_67(x, k_0), F_68(x, k_0)*F_9(x))
Eq(F_68(x, k_0), F_57(x, k_0) + F_64(x, k_0))
Eq(F_69(x, k_0), F_53(x, k_0)*F_9(x))
Eq(F_70(x, k_0), (-k_0*F_49(x, k_0) + F_49(x, 1))/(1 - k_0))
Pack JSON:
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Specification JSON:
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