0123_0231_1203_1230_3012
Counting sequence:
1, 1, 2, 6, 19, 57, 167, 499, 1556, 5072, 17126, 59281, 208727, 743901, 2675805, 9696665, 35360050, 129647850, 477642576, 1767268035, 6564126405, 24466274335, 91482572495, 343059624276, 1289904159974, 4861946416402, 18367353089702, 69533550936479, 263747951774111, 1002242216678773, 3814986502123769, 14544636039262869, 55534064877089118, 212336130412289486, 812944042149783124, 3116285494907360167, 11959798385860519537, 45950804324621816179, 176733862787006783651, 680425371729975891780, 2622127042276492210090, 10113918591637898245950, 39044429911904444082650, 150853479205085351796451, 583300119592996693237035, 2257117854077248073416905, 8740328711533173390224685, 33868773757191046886624070, 131327898242169365478203776, 509552245179617138054838872, 1978261657756160653624024356, 7684785670514316385231086881, 29869166945772625950142710337, 116157871455782434250554162131, 451959718027953471447609850479, 1759414616608818870992480243262, 6852456927844873497549658859322, 26700952856774851904245221336934, 104088460289122304033498319267206, 405944995127576985730643443854747, 1583850964596120042686772779560751, 6182127958584855650487080847774181, 24139737743045626825711458546868977, 94295850558771979787935384947015501, 368479169875816659479009042714223990, 1440418573150919668872489894244586070, 5632681584560312734993915705849911580, 22033725021956517463358552614057764335, 86218923998960285726185640663701973001, 337485502510215975556783793455059541595, 1321422108420282270489942177190230516235, 5175569924646105559418940193995066745140, 20276890389709399862928998568254642114130, 79463489365077377841208237632349270035126, 311496878311103321137536291518809135242690, 1221395654430378811828760722007962132073995, 4790408930363303911328386208394864462377795, 18793142726809884575211361279087545194676465, 73745243611532458459690151854647329240838101, 289450081175264899454283846029490767265973810, 1136359577947336271931632877004667456669277680, 4462290049988320482463241297506133183501403800, 17526585015616776834735140517915655636398071900, 68854441132780194707888052034668647142987135601, 270557451039395118028642463289168566420673305225, 1063353702922273835973036658043476458723105528075, 4180080073556524734514695828170907458428753540215, 16435314834665426797069144960762886143367592726830, 64633260585762914370496637486146181462681537702626, 254224158304000796523953440778841647086547374581790, 1000134600800354781929399250536541864362461092623470, 3935312233584004685417853572763349509774031682817955, 15487357822491889407128326963778343232013931130755335, 60960876535340415751462563580829648891969728910487501, 239993345518077005168915776623476723006280827491413145, 944973797977428207852605870454939596837230758238226010, 3721443204405954385563870541379246659709506697382159140, 14657929356129575437016877846657032761712954950903367380, 57743358069601357782187700608042856334020731624760375376, 227508830794229349661819540395688853956041682601544968565, 896519947090131496687170070074100632420837521538749992245
Generating function in Maple syntax:
1/2*(-(x-1)^5*(-4*x+1)^(1/2)+x^5-7*x^4+10*x^3-10*x^2+5*x-1)/x/(x-1)^5
Generating function in latex syntax:
\frac{-\left(x -1\right)^{5} \sqrt{-4 x +1}+x^{5}-7 x^{4}+10 x^{3}-10 x^{2}+5 x -1}{2 x \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**5/2 - 7*x**4/2 + 5*x**3 - 5*x**2 + 5*x/2 - sqrt(1 - 4*x)*(x - 1)**5/2 - 1/2)/(x*(x - 1)**5)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^10*F(x)^2-(x^5-7*x^4+10*x^3-10*x^2+5*x-1)*(x-1)^5*F(x)+x^10-10*x^9+44*x^8-114*x^7+200*x^6-242*x^5+205*x^4-119*x^3+45*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{10} F \! \left(x \right)^{2}-\left(x^{5}-7 x^{4}+10 x^{3}-10 x^{2}+5 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+x^{10}-10 x^{9}+44 x^{8}-114 x^{7}+200 x^{6}-242 x^{5}+205 x^{4}-119 x^{3}+45 x^{2}-10 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 167
a(n+1) = 2*(2*n+1)/(n+2)*a(n)-1/24*n*(n-1)*(3*n+2)*(n-4)*(n+1)/(n+2), n >= 7
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) = 19
a \! \left(5\right) = 57
a \! \left(6\right) = 167
a \! \left(n +1\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{n \left(n -1\right) \left(3 n +2\right) \left(n -4\right) \left(n +1\right)}{24 \left(n +2\right)}, \quad n \geq 7
Specification 1
Strategy pack name: requirement_placements_tracked_fusion
Tree: http://permpal.com/tree/21534/
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[41,x]+F[9,x]
F[9,x] = F[10,x]+F[19,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[5,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[5,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[3,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]+F[21,x]+F[36,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[5,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]+F[3,x]
F[24,x] = F[20,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[5,x]
F[26,x] = F[27,x]+F[3,x]
F[27,x] = F[25,x]
F[28,x] = F[29,x]*F[5,x]
F[29,x] = F[24,x]+F[3,x]
F[30,x] = F[19,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[5,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[24,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]*F[5,x]
F[37,x] = F[3,x]+F[38,x]
F[38,x] = F[20,x]+F[36,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[42,x] = F[43,x,1]
F[43,x,k[0]] = F[44,x,k[0]]
F[44,x,k[0]] = F[45,x,k[0]]
F[45,x,k[0]] = F[46,x,k[0]]*F[5,x]*F[55,x,k[0]]
F[46,x,k[0]] = F[47,x,k[0]]+F[65,x,k[0]]
F[47,x,k[0]] = F[37,x]+F[48,x,k[0]]
F[48,x,k[0]] = F[49,x,k[0]]+F[56,x,k[0]]
F[49,x,k[0]] = F[50,x,k[0]]
F[50,x,k[0]] = F[5,x]*F[51,x,k[0]]
F[51,x,k[0]] = F[49,x,k[0]]+F[52,x,k[0]]
F[52,x,k[0]] = F[53,x,k[0]]
F[53,x,k[0]] = F[54,x,k[0]]*F[55,x,k[0]]
F[54,x,k[0]] = k[0]*x
F[55,x,k[0]] = F[2,x]+F[52,x,k[0]]
F[56,x,k[0]] = 2*F[20,x]+F[57,x,k[0]]+F[64,x,k[0]]
F[57,x,k[0]] = F[5,x]*F[58,x,k[0]]
F[58,x,k[0]] = F[56,x,k[0]]+F[59,x,k[0]]
F[59,x,k[0]] = F[20,x]+F[60,x,k[0]]+F[62,x,k[0]]
F[60,x,k[0]] = F[54,x,k[0]]*F[61,x,k[0]]
F[61,x,k[0]] = F[3,x]+F[59,x,k[0]]
F[62,x,k[0]] = F[5,x]*F[63,x,k[0]]
F[63,x,k[0]] = F[52,x,k[0]]+F[59,x,k[0]]
F[64,x,k[0]] = F[48,x,k[0]]*F[5,x]
F[65,x,k[0]] = (F[44,x,k[0]]*k[0]-F[44,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_{41}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{19}\! \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_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{5}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{3}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{20}\! \left(x \right) = 0
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_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{3}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{5}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{3}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{5}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)+F_{3}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{5}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{24}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{5}\! \left(x \right)
F_{37}\! \left(x \right) = F_{3}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)+F_{36}\! \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_{42}\! \left(x \right) = F_{43}\! \left(x , 1\right)
F_{43}\! \left(x , y\right) = F_{44}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{45}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{46}\! \left(x , y\right) F_{5}\! \left(x \right) F_{55}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{47}\! \left(x , y\right)+F_{65}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{37}\! \left(x \right)+F_{48}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{49}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{50}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{51}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{49}\! \left(x , y\right)+F_{52}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{53}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{54}\! \left(x , y\right) F_{55}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = y x
F_{55}\! \left(x , y\right) = F_{2}\! \left(x \right)+F_{52}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = 2 F_{20}\! \left(x \right)+F_{57}\! \left(x , y\right)+F_{64}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{58}\! \left(x , y\right)
F_{58}\! \left(x , y\right) = F_{56}\! \left(x , y\right)+F_{59}\! \left(x , y\right)
F_{59}\! \left(x , y\right) = F_{20}\! \left(x \right)+F_{60}\! \left(x , y\right)+F_{62}\! \left(x , y\right)
F_{60}\! \left(x , y\right) = F_{54}\! \left(x , y\right) F_{61}\! \left(x , y\right)
F_{61}\! \left(x , y\right) = F_{3}\! \left(x \right)+F_{59}\! \left(x , y\right)
F_{62}\! \left(x , y\right) = F_{5}\! \left(x \right) F_{63}\! \left(x , y\right)
F_{63}\! \left(x , y\right) = F_{52}\! \left(x , y\right)+F_{59}\! \left(x , y\right)
F_{64}\! \left(x , y\right) = F_{48}\! \left(x , y\right) F_{5}\! \left(x \right)
F_{65}\! \left(x , y\right) = \frac{y F_{44}\! \left(x , y\right)-F_{44}\! \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_41(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_19(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_5(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_5(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_3(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_36(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_5(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x) + F_3(x))
Eq(F_24(x), F_20(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_5(x))
Eq(F_26(x), F_27(x) + F_3(x))
Eq(F_27(x), F_25(x))
Eq(F_28(x), F_29(x)*F_5(x))
Eq(F_29(x), F_24(x) + F_3(x))
Eq(F_30(x), F_19(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_5(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_24(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x)*F_5(x))
Eq(F_37(x), F_3(x) + F_38(x))
Eq(F_38(x), F_20(x) + F_36(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))
Eq(F_42(x), F_43(x, 1))
Eq(F_43(x, k_0), F_44(x, k_0))
Eq(F_44(x, k_0), F_45(x, k_0))
Eq(F_45(x, k_0), F_46(x, k_0)*F_5(x)*F_55(x, k_0))
Eq(F_46(x, k_0), F_47(x, k_0) + F_65(x, k_0))
Eq(F_47(x, k_0), F_37(x) + F_48(x, k_0))
Eq(F_48(x, k_0), F_49(x, k_0) + F_56(x, k_0))
Eq(F_49(x, k_0), F_50(x, k_0))
Eq(F_50(x, k_0), F_5(x)*F_51(x, k_0))
Eq(F_51(x, k_0), F_49(x, k_0) + F_52(x, k_0))
Eq(F_52(x, k_0), F_53(x, k_0))
Eq(F_53(x, k_0), F_54(x, k_0)*F_55(x, k_0))
Eq(F_54(x, k_0), k_0*x)
Eq(F_55(x, k_0), F_2(x) + F_52(x, k_0))
Eq(F_56(x, k_0), 2*F_20(x) + F_57(x, k_0) + F_64(x, k_0))
Eq(F_57(x, k_0), F_5(x)*F_58(x, k_0))
Eq(F_58(x, k_0), F_56(x, k_0) + F_59(x, k_0))
Eq(F_59(x, k_0), F_20(x) + F_60(x, k_0) + F_62(x, k_0))
Eq(F_60(x, k_0), F_54(x, k_0)*F_61(x, k_0))
Eq(F_61(x, k_0), F_3(x) + F_59(x, k_0))
Eq(F_62(x, k_0), F_5(x)*F_63(x, k_0))
Eq(F_63(x, k_0), F_52(x, k_0) + F_59(x, k_0))
Eq(F_64(x, k_0), F_48(x, k_0)*F_5(x))
Eq(F_65(x, k_0), (-k_0*F_44(x, k_0) + F_44(x, 1))/(1 - k_0))
Pack JSON:
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Specification JSON:
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