0123_0213_0231_1230_2013_3012

Counting sequence:
1, 1, 2, 6, 18, 52, 152, 464, 1486, 4946, 16916, 58951, 208232, 743186, 2674804, 9695300, 35358230, 129645470, 477639516, 1767264159, 6564121560, 24466268350, 91482565180, 343059615421, 1289904149348, 4861946403752, 18367353074752, 69533550918929, 263747951753636, 1002242216655022, 3814986502096364, 14544636039231404, 55534064877053158, 212336130412248566, 812944042149736748, 3116285494907307807, 11959798385860460632, 45950804324621750134, 176733862787006709836, 680425371729975809529, 2622127042276492118700, 10113918591637898144680, 39044429911904443970720, 150853479205085351673041, 583300119592996693101284, 2257117854077248073267910, 8740328711533173390061500, 33868773757191046886445705, 131327898242169365478009196, 509552245179617138054626996, 1978261657756160653623794056, 7684785670514316385230836981, 29869166945772625950142439612, 116157871455782434250553869306, 451959718027953471447609534228, 1759414616608818870992479902207, 6852456927844873497549658492032, 26700952856774851904245220941924, 104088460289122304033498318842936, 405944995127576985730643443399621, 1583850964596120042686772779073116, 6182127958584855650487080847252326, 24139737743045626825711458546311132, 94295850558771979787935384946419836, 368479169875816659479009042713588614, 1440418573150919668872489894243909030, 5632681584560312734993915705849190860, 22033725021956517463358552614056997855, 86218923998960285726185640663701158616, 337485502510215975556783793455058677094, 1321422108420282270489942177190229599340, 5175569924646105559418940193995065773505, 20276890389709399862928998568254641085340, 79463489365077377841208237632349268946696, 311496878311103321137536291518809134092064, 1221395654430378811828760722007962130858545, 4790408930363303911328386208394864461094820, 18793142726809884575211361279087545193323190, 73745243611532458459690151854647329239411676, 289450081175264899454283846029490767264471309, 1136359577947336271931632877004667456667696100, 4462290049988320482463241297506133183499740060, 17526585015616776834735140517915655636396322840, 68854441132780194707888052034668647142985297981, 270557451039395118028642463289168566420671375724, 1063353702922273835973036658043476458723103503290, 4180080073556524734514695828170907458428751416660, 16435314834665426797069144960762886143367590500935, 64633260585762914370496637486146181462681535370736, 254224158304000796523953440778841647086547372140164, 1000134600800354781929399250536541864362461090068280, 3935312233584004685417853572763349509774031680145285, 15487357822491889407128326963778343232013931127961180, 60960876535340415751462563580829648891969728907567766, 239993345518077005168915776623476723006280827488363644, 944973797977428207852605870454939596837230758235042465, 3721443204405954385563870541379246659709506697378837180, 14657929356129575437016877846657032761712954950899902540, 57743358069601357782187700608042856334020731624756763096, 227508830794229349661819540395688853956041682601541204189, 896519947090131496687170070074100632420837521538746071020

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

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

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

Implicit equation for the generating function in Maple syntax:
x*(x-1)^8*F(x)^2-(3*x^4-4*x^3+6*x^2-4*x+1)*(x-1)^4*F(x)+x^8-6*x^7+24*x^6-50*x^5+66*x^4-55*x^3+28*x^2-8*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{8} F \! \left(x \right)^{2}-\left(3 x^{4}-4 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{8}-6 x^{7}+24 x^{6}-50 x^{5}+66 x^{4}-55 x^{3}+28 x^{2}-8 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 52
a(1+n) = 2*(2*n+1)/(n+2)*a(n)-1/2*n*(n-1)*(n^2-3*n-2)/(n+2), n >= 6

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) = 52
a \! \left(1+n \right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{n \left(n -1\right) \left(n^{2}-3 n -2\right)}{2 \left(n +2\right)}, \quad n \geq 6

Specification 1
Strategy pack name: requirement_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22107/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[6,x]
F[1,x] = F[2,x]+F[3,x]
F[2,x] = 1
F[3,x] = F[4,x]
F[4,x] = F[1,x]*F[5,x]
F[5,x] = x
F[6,x] = F[7,x]
F[7,x] = F[5,x]*F[8,x]
F[8,x] = F[45,x]+F[9,x]
F[9,x] = F[10,x]+F[20,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[5,x]
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[2,x]+F[5,x]
F[14,x] = F[15,x]+F[3,x]
F[15,x] = F[16,x]+F[17,x]+F[19,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[5,x]
F[18,x] = F[15,x]+F[5,x]
F[19,x] = F[3,x]*F[5,x]
F[20,x] = F[16,x]+F[21,x]+F[32,x]
F[21,x] = F[22,x]*F[5,x]
F[22,x] = F[23,x]+F[28,x]
F[23,x] = F[24,x]+F[3,x]
F[24,x] = F[16,x]+F[25,x]+F[26,x]
F[25,x] = F[3,x]*F[5,x]
F[26,x] = F[27,x]*F[5,x]
F[27,x] = F[24,x]+F[5,x]
F[28,x] = F[29,x]+F[34,x]
F[29,x] = F[16,x]+F[30,x]+F[32,x]
F[30,x] = F[31,x]*F[5,x]
F[31,x] = F[29,x]+F[3,x]
F[32,x] = F[33,x]*F[5,x]
F[33,x] = F[29,x]+F[3,x]
F[34,x] = F[16,x]+F[35,x]+F[37,x]+F[41,x]
F[35,x] = F[36,x]*F[5,x]
F[36,x] = F[24,x]+F[34,x]
F[37,x] = F[38,x]*F[5,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]*F[5,x]
F[40,x] = F[3,x]+F[38,x]
F[41,x] = F[42,x]*F[5,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[19,x]
F[44,x] = 2*F[16,x]+F[37,x]+F[41,x]
F[45,x] = F[46,x]
F[46,x] = F[47,x,1]
F[47,x,k[0]] = F[48,x,k[0]]
F[48,x,k[0]] = F[49,x,k[0]]
F[49,x,k[0]] = F[5,x]*F[50,x,k[0]]*F[59,x,k[0]]
F[50,x,k[0]] = F[51,x,k[0]]+F[69,x,k[0]]
F[51,x,k[0]] = F[33,x]+F[52,x,k[0]]
F[52,x,k[0]] = F[53,x,k[0]]+F[60,x,k[0]]
F[53,x,k[0]] = F[54,x,k[0]]
F[54,x,k[0]] = F[5,x]*F[55,x,k[0]]
F[55,x,k[0]] = F[53,x,k[0]]+F[56,x,k[0]]
F[56,x,k[0]] = F[57,x,k[0]]
F[57,x,k[0]] = F[58,x,k[0]]*F[59,x,k[0]]
F[58,x,k[0]] = k[0]*x
F[59,x,k[0]] = F[2,x]+F[56,x,k[0]]
F[60,x,k[0]] = 2*F[16,x]+F[61,x,k[0]]+F[68,x,k[0]]
F[61,x,k[0]] = F[5,x]*F[62,x,k[0]]
F[62,x,k[0]] = F[60,x,k[0]]+F[63,x,k[0]]
F[63,x,k[0]] = F[16,x]+F[64,x,k[0]]+F[66,x,k[0]]
F[64,x,k[0]] = F[58,x,k[0]]*F[65,x,k[0]]
F[65,x,k[0]] = F[3,x]+F[63,x,k[0]]
F[66,x,k[0]] = F[5,x]*F[67,x,k[0]]
F[67,x,k[0]] = F[56,x,k[0]]+F[63,x,k[0]]
F[68,x,k[0]] = F[5,x]*F[52,x,k[0]]
F[69,x,k[0]] = (F[48,x,k[0]]*k[0]-F[48,x,1])/(-1+k[0])
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{1}\! \left(x \right) = F_{2}\! \left(x \right)+F_{3}\! \left(x \right)
F_{2}\! \left(x \right) = 1
F_{3}\! \left(x \right) = F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{1}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = x
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{45}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{2}\! \left(x \right)+F_{5}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{3}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{5}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{5}\! \left(x \right)
F_{19}\! \left(x \right) = F_{3}\! \left(x \right) F_{5}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{5}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{3}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{3}\! \left(x \right) F_{5}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{5}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{5}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{5}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right)+F_{3}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{5}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)+F_{3}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{5}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{5}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{5}\! \left(x \right)
F_{40}\! \left(x \right) = F_{3}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right) F_{5}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{19}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x , 1\right)
F_{47}\! \left(x , y\right) = F_{48}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{49}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{50}\! \left(x , y\right) F_{59}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{51}\! \left(x , y\right)+F_{69}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{33}\! \left(x \right)+F_{52}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{53}\! \left(x , y\right)+F_{60}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{55}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{53}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{57}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{58}\! \left(x , y\right) F_{59}\! \left(x , y\right)
F_{58}\! \left(x , y\right) = y x
F_{59}\! \left(x , y\right) = F_{2}\! \left(x \right)+F_{56}\! \left(x , y\right)
F_{60}\! \left(x , y\right) = 2 F_{16}\! \left(x \right)+F_{61}\! \left(x , y\right)+F_{68}\! \left(x , y\right)
F_{61}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{62}\! \left(x , y\right)
F_{62}\! \left(x , y\right) = F_{60}\! \left(x , y\right)+F_{63}\! \left(x , y\right)
F_{63}\! \left(x , y\right) = F_{16}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{66}\! \left(x , y\right)
F_{64}\! \left(x , y\right) = F_{58}\! \left(x , y\right) F_{65}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = F_{3}\! \left(x \right)+F_{63}\! \left(x , y\right)
F_{66}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{67}\! \left(x , y\right)
F_{67}\! \left(x , y\right) = F_{56}\! \left(x , y\right)+F_{63}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{52}\! \left(x , y\right)
F_{69}\! \left(x , y\right) = \frac{y F_{48}\! \left(x , y\right)-F_{48}\! \left(x , 1\right)}{-1+y}
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_6(x))
Eq(F_1(x), F_2(x) + F_3(x))
Eq(F_2(x), 1)
Eq(F_3(x), F_4(x))
Eq(F_4(x), F_1(x)*F_5(x))
Eq(F_5(x), x)
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)*F_8(x))
Eq(F_8(x), F_45(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_20(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_5(x))
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_2(x) + F_5(x))
Eq(F_14(x), F_15(x) + F_3(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_19(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_5(x))
Eq(F_18(x), F_15(x) + F_5(x))
Eq(F_19(x), F_3(x)*F_5(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_32(x))
Eq(F_21(x), F_22(x)*F_5(x))
Eq(F_22(x), F_23(x) + F_28(x))
Eq(F_23(x), F_24(x) + F_3(x))
Eq(F_24(x), F_16(x) + F_25(x) + F_26(x))
Eq(F_25(x), F_3(x)*F_5(x))
Eq(F_26(x), F_27(x)*F_5(x))
Eq(F_27(x), F_24(x) + F_5(x))
Eq(F_28(x), F_29(x) + F_34(x))
Eq(F_29(x), F_16(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_31(x)*F_5(x))
Eq(F_31(x), F_29(x) + F_3(x))
Eq(F_32(x), F_33(x)*F_5(x))
Eq(F_33(x), F_29(x) + F_3(x))
Eq(F_34(x), F_16(x) + F_35(x) + F_37(x) + F_41(x))
Eq(F_35(x), F_36(x)*F_5(x))
Eq(F_36(x), F_24(x) + F_34(x))
Eq(F_37(x), F_38(x)*F_5(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x)*F_5(x))
Eq(F_40(x), F_3(x) + F_38(x))
Eq(F_41(x), F_42(x)*F_5(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_19(x))
Eq(F_44(x), 2*F_16(x) + F_37(x) + F_41(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(x, 1))
Eq(F_47(x, k_0), F_48(x, k_0))
Eq(F_48(x, k_0), F_49(x, k_0))
Eq(F_49(x, k_0), F_5(x)*F_50(x, k_0)*F_59(x, k_0))
Eq(F_50(x, k_0), F_51(x, k_0) + F_69(x, k_0))
Eq(F_51(x, k_0), F_33(x) + F_52(x, k_0))
Eq(F_52(x, k_0), F_53(x, k_0) + F_60(x, k_0))
Eq(F_53(x, k_0), F_54(x, k_0))
Eq(F_54(x, k_0), F_5(x)*F_55(x, k_0))
Eq(F_55(x, k_0), F_53(x, k_0) + F_56(x, k_0))
Eq(F_56(x, k_0), F_57(x, k_0))
Eq(F_57(x, k_0), F_58(x, k_0)*F_59(x, k_0))
Eq(F_58(x, k_0), k_0*x)
Eq(F_59(x, k_0), F_2(x) + F_56(x, k_0))
Eq(F_60(x, k_0), 2*F_16(x) + F_61(x, k_0) + F_68(x, k_0))
Eq(F_61(x, k_0), F_5(x)*F_62(x, k_0))
Eq(F_62(x, k_0), F_60(x, k_0) + F_63(x, k_0))
Eq(F_63(x, k_0), F_16(x) + F_64(x, k_0) + F_66(x, k_0))
Eq(F_64(x, k_0), F_58(x, k_0)*F_65(x, k_0))
Eq(F_65(x, k_0), F_3(x) + F_63(x, k_0))
Eq(F_66(x, k_0), F_5(x)*F_67(x, k_0))
Eq(F_67(x, k_0), F_56(x, k_0) + F_63(x, k_0))
Eq(F_68(x, k_0), F_5(x)*F_52(x, k_0))
Eq(F_69(x, k_0), (-k_0*F_48(x, k_0) + F_48(x, 1))/(1 - k_0))
Pack JSON:
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Specification JSON:
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