0123_0213_0231_0312_1203_3012
Counting sequence:
1, 1, 2, 6, 18, 53, 159, 495, 1600, 5333, 18194, 63167, 222294, 790797, 2838452, 10265484, 37368348, 136803359, 503347857, 1860283302, 6902856644, 25706763213, 96048126849, 359937671342, 1352548393986, 5095293095211, 19239398084785, 72802194266612, 276033039859496, 1048530949011737, 3989799629112615, 15206242263066231, 58042975116516968, 221867842088244069, 849218688722393562, 3254557274713803747, 12487657627650216622, 47968785399293917861, 184458785560367742248, 710033854453692613065, 2735746507526853342162, 10550410324778199138269, 40723080570027955885026, 157315702487561837767473, 608201247790833814769270, 2353157649393357935862117, 9111060415375553450595294, 35301050730560691295020491, 136865691700144516457980098, 530979855048809211219395457, 2061232157720372111473191008, 8006280032499519055742685255, 31115718386884408799349170610, 120994274850882546898111688885, 470735553922417979898405464400, 1832348632493796467201543382495, 7135925742667295307722572807838, 27803292807509602526035178888073, 108377422329353086935550993090224, 422640787138732377879154165083155, 1648874747316403472952182100240666, 6435489353041592851773327638763741, 25127391365393720516653125122393372, 98147601989762452729853826400338756, 383506952654785390083712694985826404, 1499074058560248162377498651821221191, 5861712294441154476424684848164713017, 22928357648025842597219546463257412582, 89714803806085984065853813436177013836, 351150956347826249650630015073049665829, 1374859188159761840349695870190032364561, 5384599606157799430782516421576678066682, 21094819151373869017091096187073119282642, 82665045793695465529953497676249898968891, 324032360918121976172068295215632267576269, 1270492252116954382883212049153802760467576, 4982757321395670124918097676418166848375992, 19546930467765378478989209996935562733851057, 76700053913418795913137775653275277459551811, 301035906179453100228211268948248726151613804, 1181799552522007240372061484361993051402481086, 4640552552613187581375882626475111368502143267, 18226088505946818326379989951698862133350712227, 71599963648417725126456737833473325130243655446, 281336060667930772390789855010708092260085337904, 1105679052766915026130152765763379864412920941101, 4346320257853259067026058713999472423110205407117, 17088395398689636996522675150099578912602585401492, 67199460777712256997453292162433944740755078943366, 264309837445629452873488872953694331036173223461159, 1039781433048082360093658258182336205289720598608871, 4091195153680013707859477591338608425447687730637702, 16100375923482363192833910869520431287289293977164252, 63372063511071892401348101500512472516064569902191121, 249479047295980027469315116256637538138186949655140709, 982297718307425207996981186464894002647953895728542266, 3868329923573308211497146572010139904262022313261066742, 15236096799937625941852895106776486816076519335721098487, 60019497961830873026892905482973432766101831827302066101, 236471078063067052092924249812132360993456312367974449224, 931814366019893684188221478805415911052914629566130848156
Generating function in Maple syntax:
1/2/x*(-1+(1-4*x)^(1/2))*(3*x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
\frac{\left(-1+\sqrt{1-4 x}\right) \left(3 x^{2}-3 x +1\right)}{2 x \left(x -1\right)^{3}}
Generating function in sympy syntax:
(sqrt(1 - 4*x) - 1)*(3*x**2 - 3*x + 1)/(2*x*(x - 1)**3)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^6*F(x)^2+(3*x^2-3*x+1)*(x-1)^3*F(x)+(3*x^2-3*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{6} F \! \left(x \right)^{2}+\left(3 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(3 x^{2}-3 x +1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(n+4) = -6*(2*n+3)/(5+n)*a(n)+9*(3*n+5)/(5+n)*a(1+n)-2*(11*n+29)/(5+n)*a(n+2)+2*(4*n+15)/(5+n)*a(n+3)+(n-1)/(5+n), n >= 5
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(n +4\right) = -\frac{6 \left(2 n +3\right) a \! \left(n \right)}{5+n}+\frac{9 \left(3 n +5\right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(11 n +29\right) a \! \left(n +2\right)}{5+n}+\frac{2 \left(4 n +15\right) a \! \left(n +3\right)}{5+n}+\frac{n -1}{5+n}, \quad n \geq 5
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22100/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[19,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[62,x]
F[4,x] = F[19,x]*F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[59,x,k[0]]+F[61,x,k[0]]+F[7,x,k[0]]
F[7,x,k[0]] = F[19,x]*F[8,x,k[0]]
F[8,x,k[0]] = F[27,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[22,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]+F[15,x,k[0]]
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]
F[13,x,k[0]] = F[11,x,k[0]]*F[14,x,k[0]]
F[14,x,k[0]] = k[0]*x
F[15,x,k[0]] = F[16,x]+F[20,x,k[0]]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[19,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = x
F[20,x,k[0]] = F[21,x,k[0]]
F[21,x,k[0]] = F[14,x,k[0]]*F[15,x,k[0]]
F[22,x,k[0]] = F[23,x,k[0]]+F[24,x,k[0]]
F[23,x,k[0]] = F[19,x]
F[24,x,k[0]] = F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[16,x]*F[19,x]
F[27,x,k[0]] = F[28,x,k[0]]+F[45,x,k[0]]
F[28,x,k[0]] = F[29,x,k[0]]+F[35,x,k[0]]
F[29,x,k[0]] = F[16,x]+F[30,x,k[0]]
F[30,x,k[0]] = F[31,x]+F[32,x,k[0]]+F[34,x,k[0]]
F[31,x] = 0
F[32,x,k[0]] = F[19,x]*F[33,x,k[0]]
F[33,x,k[0]] = F[12,x,k[0]]+F[30,x,k[0]]
F[34,x,k[0]] = F[14,x,k[0]]*F[29,x,k[0]]
F[35,x,k[0]] = F[36,x]+F[41,x,k[0]]
F[36,x] = F[31,x]+F[37,x]+F[39,x]
F[37,x] = F[19,x]*F[38,x]
F[38,x] = F[16,x]+F[36,x]
F[39,x] = F[19,x]*F[40,x]
F[40,x] = F[16,x]+F[36,x]
F[41,x,k[0]] = 2*F[31,x]+F[42,x,k[0]]+F[44,x,k[0]]
F[42,x,k[0]] = F[19,x]*F[43,x,k[0]]
F[43,x,k[0]] = F[20,x,k[0]]+F[41,x,k[0]]
F[44,x,k[0]] = F[14,x,k[0]]*F[35,x,k[0]]
F[45,x,k[0]] = F[46,x,k[0]]+F[51,x,k[0]]
F[46,x,k[0]] = F[47,x]
F[47,x] = F[31,x]+F[48,x]+F[50,x]
F[48,x] = F[19,x]*F[49,x]
F[49,x] = F[19,x]+F[47,x]
F[50,x] = F[16,x]*F[19,x]
F[51,x,k[0]] = F[52,x]
F[52,x] = 2*F[31,x]+F[53,x]+F[55,x]
F[53,x] = F[19,x]*F[54,x]
F[54,x] = F[25,x]+F[52,x]
F[55,x] = F[19,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[19,x]*F[58,x]
F[58,x] = F[16,x]+F[56,x]
F[59,x,k[0]] = F[19,x]*F[60,x,k[0]]
F[60,x,k[0]] = 1/(-1+k[0])*(F[6,x,k[0]]*k[0]-F[6,x,1])
F[61,x,k[0]] = F[14,x,k[0]]*F[6,x,k[0]]
F[62,x] = F[19,x]*F[63,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = F[18,x]+F[65,x]
F[65,x] = F[19,x]+F[25,x]
F[66,x] = F[40,x]+F[67,x]
F[67,x] = F[47,x]+F[52,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_{19}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{62}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{59}\! \left(x , y\right)+F_{61}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{19}\! \left(x \right) F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{27}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{22}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = y x
F_{15}\! \left(x , y\right) = F_{16}\! \left(x \right)+F_{20}\! \left(x , y\right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = x
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{19}\! \left(x \right)
F_{24}\! \left(x , y\right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{27}\! \left(x , y\right) = F_{28}\! \left(x , y\right)+F_{45}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right)+F_{35}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{16}\! \left(x \right)+F_{30}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x \right)+F_{32}\! \left(x , y\right)+F_{34}\! \left(x , y\right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x , y\right) = F_{19}\! \left(x \right) F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{12}\! \left(x , y\right)+F_{30}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{29}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{41}\! \left(x , y\right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{19}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \left(x \right)
F_{41}\! \left(x , y\right) = 2 F_{31}\! \left(x \right)+F_{42}\! \left(x , y\right)+F_{44}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{19}\! \left(x \right) F_{43}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{20}\! \left(x , y\right)+F_{41}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{35}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{46}\! \left(x , y\right)+F_{51}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{31}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{19}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{51}\! \left(x , y\right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{19}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{25}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{19}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{19}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{16}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x , y\right) = F_{19}\! \left(x \right) F_{60}\! \left(x , y\right)
F_{60}\! \left(x , y\right) = \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}
F_{61}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{6}\! \left(x , y\right)
F_{62}\! \left(x \right) = F_{19}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{18}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{19}\! \left(x \right)+F_{25}\! \left(x \right)
F_{66}\! \left(x \right) = F_{40}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{47}\! \left(x \right)+F_{52}\! \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_19(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_62(x))
Eq(F_4(x), F_19(x)*F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_59(x, k_0) + F_61(x, k_0) + F_7(x, k_0))
Eq(F_7(x, k_0), F_19(x)*F_8(x, k_0))
Eq(F_8(x, k_0), F_27(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_22(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0) + F_15(x, k_0))
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0))
Eq(F_13(x, k_0), F_11(x, k_0)*F_14(x, k_0))
Eq(F_14(x, k_0), k_0*x)
Eq(F_15(x, k_0), F_16(x) + F_20(x, k_0))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_19(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), x)
Eq(F_20(x, k_0), F_21(x, k_0))
Eq(F_21(x, k_0), F_14(x, k_0)*F_15(x, k_0))
Eq(F_22(x, k_0), F_23(x, k_0) + F_24(x, k_0))
Eq(F_23(x, k_0), F_19(x))
Eq(F_24(x, k_0), F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_16(x)*F_19(x))
Eq(F_27(x, k_0), F_28(x, k_0) + F_45(x, k_0))
Eq(F_28(x, k_0), F_29(x, k_0) + F_35(x, k_0))
Eq(F_29(x, k_0), F_16(x) + F_30(x, k_0))
Eq(F_30(x, k_0), F_31(x) + F_32(x, k_0) + F_34(x, k_0))
Eq(F_31(x), 0)
Eq(F_32(x, k_0), F_19(x)*F_33(x, k_0))
Eq(F_33(x, k_0), F_12(x, k_0) + F_30(x, k_0))
Eq(F_34(x, k_0), F_14(x, k_0)*F_29(x, k_0))
Eq(F_35(x, k_0), F_36(x) + F_41(x, k_0))
Eq(F_36(x), F_31(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_19(x)*F_38(x))
Eq(F_38(x), F_16(x) + F_36(x))
Eq(F_39(x), F_19(x)*F_40(x))
Eq(F_40(x), F_16(x) + F_36(x))
Eq(F_41(x, k_0), 2*F_31(x) + F_42(x, k_0) + F_44(x, k_0))
Eq(F_42(x, k_0), F_19(x)*F_43(x, k_0))
Eq(F_43(x, k_0), F_20(x, k_0) + F_41(x, k_0))
Eq(F_44(x, k_0), F_14(x, k_0)*F_35(x, k_0))
Eq(F_45(x, k_0), F_46(x, k_0) + F_51(x, k_0))
Eq(F_46(x, k_0), F_47(x))
Eq(F_47(x), F_31(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_19(x)*F_49(x))
Eq(F_49(x), F_19(x) + F_47(x))
Eq(F_50(x), F_16(x)*F_19(x))
Eq(F_51(x, k_0), F_52(x))
Eq(F_52(x), 2*F_31(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_19(x)*F_54(x))
Eq(F_54(x), F_25(x) + F_52(x))
Eq(F_55(x), F_19(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_19(x)*F_58(x))
Eq(F_58(x), F_16(x) + F_56(x))
Eq(F_59(x, k_0), F_19(x)*F_60(x, k_0))
Eq(F_60(x, k_0), (k_0*F_6(x, k_0) - F_6(x, 1))/(k_0 - 1))
Eq(F_61(x, k_0), F_14(x, k_0)*F_6(x, k_0))
Eq(F_62(x), F_19(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), F_18(x) + F_65(x))
Eq(F_65(x), F_19(x) + F_25(x))
Eq(F_66(x), F_40(x) + F_67(x))
Eq(F_67(x), F_47(x) + F_52(x))
Pack JSON:
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Specification JSON:
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