0132_0231_1203_3012
Counting sequence:
1, 1, 2, 6, 20, 64, 196, 597, 1861, 6000, 19973, 68213, 237500, 838948, 2996679, 10799632, 39207786, 143234792, 526107106, 1941619434, 7195942157, 26770341286, 99931433532, 374192600636, 1405125001873, 5290036254114, 19963478877868, 75503711778720, 286143889085453, 1086480617806574, 4132611850737326, 15744969471091933, 60079727899241625, 229584021573745588, 878507201799365769, 3365926792612102509, 12911845351734183152, 49586963386763913136, 190640784938971851647, 733683680453690979657, 2826338221747646374952, 10897849452223431450418, 42057122629004151439015, 162443537595925358177283, 627932180122273819318412, 2429153061402948669880492, 9404036546215201031511693, 36431530064401208859643235, 141231456684566702900333818, 547853483922551344030316574, 2126498799790473541593674093, 8258915284367235474726788329, 32094312862686742017206428434, 124787466570893648484894322004, 485448103873741089629510536627, 1889449257629148706481798207679, 7357670235467067247871750601044, 28664908945839195682739248874854, 111727187183974603146007893371349, 435670816024550210515506371895977, 1699585322264762872787608424763718, 6632943285441674504288083327406376, 25896591869427560002005783715332587, 101145471132949216798376064937065496, 395195978282301193727013483006590466, 1544670401176691900172125185840502792, 6039647390655223931834032549610985754, 23623009649410924028995441206912176994, 92427751769490557506782465020548271429, 361750296551551544904756006051573196726, 1416285315845833382131182477177013883052, 5546565347160440386377356636181256266224, 21728281335139377285272842549840725002809, 85143395182301605977823015504236073553274, 333731751925628761151590430750238180586376, 1308464164767116565620543452125955570660132, 5131457912026002104798863737378824051470689, 20129424438147018457103850321088665341325254, 78982471756743789912397824881147717535579514, 309981732367506263038144414491225080208674946, 1216871839020323142183297099541113271454201587, 4778090613265033620841902879287640736910819634, 18765591546423069532111721469806673466499232794, 73716736172178926414390809548116527642303620434, 289643381713297076007506147756743857243616679531, 1138289115584990168889732527751299359062538847002, 4474359629989152244462018083148738254978872443644, 17591240489263640029138476398625878079523978160194, 69174707180631414761487516675462194410827915807969, 272070564796471506090889937008634800031628142697046, 1070279718874131068637086841263269805462500764396242, 4211072783143146229080959621885983391424612765175774, 16571666225385241337102899944864118425540790621588199, 65225272935051727139469882405915594680540448295886274, 256767648772413289292913556907148395606127872887670756, 1010968910315395991071263844301409314003801904289647652, 3981134658019176017160579201499724842371222718976359669, 15679998893048198399527255113537812649101474110694774514, 61766621661724900320172576040604559952406430987800801344, 243348648991228160380450347379744095098286676318385438056, 958892571377959712582662200080845235664655353251954095857
Generating function in Maple syntax:
-1/2*(-1+(1-4*x)^(1/2))/x*(x^5+2*x^4-3*x^3+6*x^2-4*x+1)/(x-1)^4
Generating function in latex syntax:
-\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{5}+2 x^{4}-3 x^{3}+6 x^{2}-4 x +1\right)}{2 x \left(x -1\right)^{4}}
Generating function in sympy syntax:
(1/2 - sqrt(1 - 4*x)/2)*(x**5 + 2*x**4 - 3*x**3 + 6*x**2 - 4*x + 1)/(x*(x - 1)**4)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^8*F(x)^2-(x^5+2*x^4-3*x^3+6*x^2-4*x+1)*(x-1)^4*F(x)+(x^5+2*x^4-3*x^3+6*x^2-4*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{8} F \! \left(x \right)^{2}-\left(x^{5}+2 x^{4}-3 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+\left(x^{5}+2 x^{4}-3 x^{3}+6 x^{2}-4 x +1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 196
a(7) = 597
a(8) = 1861
a(9) = 6000
a(10) = 19973
a(n+7) = -2*(2*n-1)/(n+8)*a(n)-(3*n+25)/(n+8)*a(n+1)+(38+21*n)/(n+8)*a(n+2)-(41*n+153)/(n+8)*a(n+3)+(228+49*n)/(n+8)*a(n+4)-10*(3*n+17)/(n+8)*a(n+5)+(61+9*n)/(n+8)*a(n+6)+3/2*(3*n^2+3*n+4)/(n+8), n >= 11
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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 196
a \! \left(7\right) = 597
a \! \left(8\right) = 1861
a \! \left(9\right) = 6000
a \! \left(10\right) = 19973
a \! \left(n +7\right) = -\frac{2 \left(2 n -1\right) a \! \left(n \right)}{n +8}-\frac{\left(3 n +25\right) a \! \left(n +1\right)}{n +8}+\frac{\left(38+21 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(41 n +153\right) a \! \left(n +3\right)}{n +8}+\frac{\left(228+49 n \right) a \! \left(n +4\right)}{n +8}-\frac{10 \left(3 n +17\right) a \! \left(n +5\right)}{n +8}+\frac{\left(61+9 n \right) a \! \left(n +6\right)}{n +8}+\frac{\frac{9}{2} n^{2}+\frac{9}{2} n +6}{n +8}, \quad n \geq 11
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21232/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[12,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[75,x]
F[4,x] = F[12,x]*F[5,x]
F[5,x] = F[38,x]+F[6,x]
F[6,x] = F[22,x]+F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[12,x]
F[10,x] = F[11,x]+F[13,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[12,x]+F[17,x]
F[21,x] = F[12,x]*F[14,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[14,x]+F[30,x]
F[30,x] = F[18,x]+F[31,x]+F[33,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[27,x]+F[30,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[14,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[27,x]+F[35,x]
F[38,x] = F[39,x]+F[62,x]
F[39,x] = F[14,x]+F[40,x]
F[40,x] = F[18,x]+F[41,x]+F[43,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[40,x]+F[8,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[49,x]
F[45,x] = F[14,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[12,x]+F[46,x]
F[49,x] = F[50,x]+F[55,x]
F[50,x] = F[18,x]+F[51,x]+F[53,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[14,x]+F[50,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[14,x]+F[50,x]
F[55,x] = 2*F[18,x]+F[56,x]+F[60,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[21,x]
F[59,x] = F[56,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[46,x]+F[55,x]
F[62,x] = F[63,x]
F[63,x] = F[18,x]+F[64,x]+F[66,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[23,x]+F[63,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[34,x]+F[68,x]
F[68,x] = F[69,x]+F[72,x]
F[69,x] = F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[14,x]+F[69,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[35,x]+F[72,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[77,x,1]
F[77,x,k[0]] = F[1,x]+F[109,x,k[0]]+F[111,x,k[0]]+F[78,x,k[0]]
F[78,x,k[0]] = F[12,x]*F[79,x,k[0]]
F[79,x,k[0]] = F[80,x,k[0]]+F[92,x,k[0]]
F[80,x,k[0]] = F[81,x,k[0]]+F[91,x,k[0]]
F[81,x,k[0]] = F[82,x,k[0]]+F[86,x,k[0]]
F[82,x,k[0]] = F[1,x]+F[83,x,k[0]]
F[83,x,k[0]] = F[84,x,k[0]]
F[84,x,k[0]] = F[82,x,k[0]]*F[85,x,k[0]]
F[85,x,k[0]] = k[0]*x
F[86,x,k[0]] = F[8,x]+F[87,x,k[0]]
F[87,x,k[0]] = F[88,x,k[0]]
F[88,x,k[0]] = F[85,x,k[0]]*F[89,x,k[0]]
F[89,x,k[0]] = F[14,x]+F[90,x,k[0]]
F[90,x,k[0]] = F[88,x,k[0]]
F[91,x,k[0]] = F[23,x]
F[92,x,k[0]] = F[108,x,k[0]]+F[93,x,k[0]]
F[93,x,k[0]] = F[94,x,k[0]]+F[99,x,k[0]]
F[94,x,k[0]] = F[14,x]+F[95,x,k[0]]
F[95,x,k[0]] = F[18,x]+F[96,x,k[0]]+F[98,x,k[0]]
F[96,x,k[0]] = F[12,x]*F[97,x,k[0]]
F[97,x,k[0]] = F[83,x,k[0]]+F[95,x,k[0]]
F[98,x,k[0]] = F[85,x,k[0]]*F[94,x,k[0]]
F[99,x,k[0]] = F[100,x,k[0]]+F[40,x]
F[100,x,k[0]] = 2*F[18,x]+F[101,x,k[0]]+F[103,x,k[0]]
F[101,x,k[0]] = F[102,x,k[0]]*F[12,x]
F[102,x,k[0]] = F[100,x,k[0]]+F[87,x,k[0]]
F[103,x,k[0]] = F[104,x,k[0]]*F[85,x,k[0]]
F[104,x,k[0]] = F[105,x,k[0]]+F[50,x]
F[105,x,k[0]] = 2*F[18,x]+F[103,x,k[0]]+F[106,x,k[0]]
F[106,x,k[0]] = F[107,x,k[0]]*F[12,x]
F[107,x,k[0]] = F[105,x,k[0]]+F[90,x,k[0]]
F[108,x,k[0]] = F[63,x]
F[109,x,k[0]] = F[110,x,k[0]]*F[12,x]
F[110,x,k[0]] = 1/(-1+k[0])*(F[77,x,k[0]]*k[0]-F[77,x,1])
F[111,x,k[0]] = F[77,x,k[0]]*F[85,x,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_{12}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{75}\! \left(x \right)
F_{4}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{38}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{13}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{62}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)+F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right)+F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{46}\! \left(x \right)+F_{55}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{23}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{34}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{72}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{14}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{35}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x , 1\right)
F_{77}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{109}\! \left(x , y\right)+F_{111}\! \left(x , y\right)+F_{78}\! \left(x , y\right)
F_{78}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{79}\! \left(x , y\right)
F_{79}\! \left(x , y\right) = F_{80}\! \left(x , y\right)+F_{92}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{81}\! \left(x , y\right)+F_{91}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{82}\! \left(x , y\right)+F_{86}\! \left(x , y\right)
F_{82}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{83}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = F_{82}\! \left(x , y\right) F_{85}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = y x
F_{86}\! \left(x , y\right) = F_{8}\! \left(x \right)+F_{87}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{88}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{85}\! \left(x , y\right) F_{89}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{14}\! \left(x \right)+F_{90}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{88}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{23}\! \left(x \right)
F_{92}\! \left(x , y\right) = F_{108}\! \left(x , y\right)+F_{93}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = F_{94}\! \left(x , y\right)+F_{99}\! \left(x , y\right)
F_{94}\! \left(x , y\right) = F_{14}\! \left(x \right)+F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = F_{18}\! \left(x \right)+F_{96}\! \left(x , y\right)+F_{98}\! \left(x , y\right)
F_{96}\! \left(x , y\right) = F_{12}\! \left(x \right) F_{97}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{83}\! \left(x , y\right)+F_{95}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{85}\! \left(x , y\right) F_{94}\! \left(x , y\right)
F_{99}\! \left(x , y\right) = F_{100}\! \left(x , y\right)+F_{40}\! \left(x \right)
F_{100}\! \left(x , y\right) = 2 F_{18}\! \left(x \right)+F_{101}\! \left(x , y\right)+F_{103}\! \left(x , y\right)
F_{101}\! \left(x , y\right) = F_{102}\! \left(x , y\right) F_{12}\! \left(x \right)
F_{102}\! \left(x , y\right) = F_{100}\! \left(x , y\right)+F_{87}\! \left(x , y\right)
F_{103}\! \left(x , y\right) = F_{104}\! \left(x , y\right) F_{85}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = F_{105}\! \left(x , y\right)+F_{50}\! \left(x \right)
F_{105}\! \left(x , y\right) = 2 F_{18}\! \left(x \right)+F_{103}\! \left(x , y\right)+F_{106}\! \left(x , y\right)
F_{106}\! \left(x , y\right) = F_{107}\! \left(x , y\right) F_{12}\! \left(x \right)
F_{107}\! \left(x , y\right) = F_{105}\! \left(x , y\right)+F_{90}\! \left(x , y\right)
F_{108}\! \left(x , y\right) = F_{63}\! \left(x \right)
F_{109}\! \left(x , y\right) = F_{110}\! \left(x , y\right) F_{12}\! \left(x \right)
F_{110}\! \left(x , y\right) = \frac{F_{77}\! \left(x , y\right) y -F_{77}\! \left(x , 1\right)}{-1+y}
F_{111}\! \left(x , y\right) = F_{77}\! \left(x , y\right) F_{85}\! \left(x , y\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_12(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_75(x))
Eq(F_4(x), F_12(x)*F_5(x))
Eq(F_5(x), F_38(x) + F_6(x))
Eq(F_6(x), F_22(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_12(x))
Eq(F_10(x), F_11(x) + F_13(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_12(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_14(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_14(x) + F_30(x))
Eq(F_30(x), F_18(x) + F_31(x) + F_33(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_27(x) + F_30(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_14(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_27(x) + F_35(x))
Eq(F_38(x), F_39(x) + F_62(x))
Eq(F_39(x), F_14(x) + F_40(x))
Eq(F_40(x), F_18(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_40(x) + F_8(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_49(x))
Eq(F_45(x), F_14(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_12(x) + F_46(x))
Eq(F_49(x), F_50(x) + F_55(x))
Eq(F_50(x), F_18(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_14(x) + F_50(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_14(x) + F_50(x))
Eq(F_55(x), 2*F_18(x) + F_56(x) + F_60(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_21(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_46(x) + F_55(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_18(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_23(x) + F_63(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_34(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_72(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_14(x) + F_69(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_35(x) + F_72(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_77(x, 1))
Eq(F_77(x, k_0), F_1(x) + F_109(x, k_0) + F_111(x, k_0) + F_78(x, k_0))
Eq(F_78(x, k_0), F_12(x)*F_79(x, k_0))
Eq(F_79(x, k_0), F_80(x, k_0) + F_92(x, k_0))
Eq(F_80(x, k_0), F_81(x, k_0) + F_91(x, k_0))
Eq(F_81(x, k_0), F_82(x, k_0) + F_86(x, k_0))
Eq(F_82(x, k_0), F_1(x) + F_83(x, k_0))
Eq(F_83(x, k_0), F_84(x, k_0))
Eq(F_84(x, k_0), F_82(x, k_0)*F_85(x, k_0))
Eq(F_85(x, k_0), k_0*x)
Eq(F_86(x, k_0), F_8(x) + F_87(x, k_0))
Eq(F_87(x, k_0), F_88(x, k_0))
Eq(F_88(x, k_0), F_85(x, k_0)*F_89(x, k_0))
Eq(F_89(x, k_0), F_14(x) + F_90(x, k_0))
Eq(F_90(x, k_0), F_88(x, k_0))
Eq(F_91(x, k_0), F_23(x))
Eq(F_92(x, k_0), F_108(x, k_0) + F_93(x, k_0))
Eq(F_93(x, k_0), F_94(x, k_0) + F_99(x, k_0))
Eq(F_94(x, k_0), F_14(x) + F_95(x, k_0))
Eq(F_95(x, k_0), F_18(x) + F_96(x, k_0) + F_98(x, k_0))
Eq(F_96(x, k_0), F_12(x)*F_97(x, k_0))
Eq(F_97(x, k_0), F_83(x, k_0) + F_95(x, k_0))
Eq(F_98(x, k_0), F_85(x, k_0)*F_94(x, k_0))
Eq(F_99(x, k_0), F_100(x, k_0) + F_40(x))
Eq(F_100(x, k_0), F_101(x, k_0) + F_103(x, k_0) + 2*F_18(x))
Eq(F_101(x, k_0), F_102(x, k_0)*F_12(x))
Eq(F_102(x, k_0), F_100(x, k_0) + F_87(x, k_0))
Eq(F_103(x, k_0), F_104(x, k_0)*F_85(x, k_0))
Eq(F_104(x, k_0), F_105(x, k_0) + F_50(x))
Eq(F_105(x, k_0), F_103(x, k_0) + F_106(x, k_0) + 2*F_18(x))
Eq(F_106(x, k_0), F_107(x, k_0)*F_12(x))
Eq(F_107(x, k_0), F_105(x, k_0) + F_90(x, k_0))
Eq(F_108(x, k_0), F_63(x))
Eq(F_109(x, k_0), F_110(x, k_0)*F_12(x))
Eq(F_110(x, k_0), (k_0*F_77(x, k_0) - F_77(x, 1))/(k_0 - 1))
Eq(F_111(x, k_0), F_77(x, k_0)*F_85(x, k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 1, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "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": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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