0132_1320_2130_3021
Counting sequence:
1, 1, 2, 6, 20, 63, 182, 492, 1279, 3268, 8323, 21278, 54729, 141578, 367872, 958775, 2503644, 6545284, 17122636, 44809585, 117288695, 307033605, 803782481, 2104275975, 5508997649, 14422657287, 37758900406, 98853953476, 258802850068, 677554464077, 1773860383242, 4644026496482, 12158218882383, 31830629887322, 83333670471351, 218170381167724, 571177472615593, 1495362036198568, 3914908635427700, 10249363869451869, 26833182972205964, 70250185046345034, 183917372165898560, 481501931450299119, 1260588422183814103, 3300263335099812211, 8640201583114131205, 22620341414240914685, 59220822659606754657, 155042126564577282461, 405905557034122798986, 1062674544537788574386, 2782118076579240117012, 7283679685199928680491, 19068920979020542516030, 49923083251861695122248, 130700328776564538742367, 342177903077831916605952, 895833380456931206156939, 2345322238292961696495978, 6140133334421953877479433, 16075077764972899929574038, 42085099960496745904321864, 110180222116517337775880563, 288455566389055267415179132, 755186477050648464460844960, 1977103864762890125957829204, 5176125117238021913402355869, 13551271486951175614238143671, 35477689343615504929300122545, 92881796543895339173649361305, 243167700288070512591634134115, 636621304320316198601238192241, 1666696212672878083212064512835, 4363467333698318051034938273534, 11423705788422076069892732027472, 29907650031567910158643238253716, 78299244306281654406036961833561, 204990082887277053059467624928978, 536671004355549504772365889141662, 1405022930179371461257630017111727, 3678397786182564879000524135154742, 9630170428368323175743942359574143, 25212113498922404648231302912961432, 66006170068398890768949966346784361, 172806396706274267658618596092851284, 452413020050423912206905821895116028, 1184432663444997468962098869553628153, 3100884970284568494679390786724578860, 8118222247408708015076073490576488454, 21253781771941555550548829684958722824, 55643123068415958636570415564250855419, 145675587433306320359162417007742236695, 381383639231503002440916835458921340479, 998475330261202686963588089368964233613, 2614042351552105058449847432647910638521, 6843651724395112488385954208574703651265, 17916912821633232406708015193076132833145, 46907086740504584731738091370653623767442, 122804347399880521788506258918884663638046, 321505955459136980633780685386000288408596
Generating function in Maple syntax:
-(x^7-12*x^6+36*x^5-53*x^4+49*x^3-27*x^2+8*x-1)/(x^2-3*x+1)/(x-1)^6
Generating function in latex syntax:
-\frac{x^{7}-12 x^{6}+36 x^{5}-53 x^{4}+49 x^{3}-27 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-x**7 + 12*x**6 - 36*x**5 + 53*x**4 - 49*x**3 + 27*x**2 - 8*x + 1)/((x - 1)**6*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^6*F(x)+x^7-12*x^6+36*x^5-53*x^4+49*x^3-27*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{6} F \! \left(x \right)+x^{7}-12 x^{6}+36 x^{5}-53 x^{4}+49 x^{3}-27 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
1/120*(108*5^(1/2)-180)*(3/2-1/2*5^(1/2))^(-n)+1/120*(-108*5^(1/2)-180)*(3/2+1/2*5^(1/2))^(-n)+1/120*n^5-3/8*n^3+3/2*n^2-62/15*n+4
Explicit closed form in latex syntax:
\frac{\left(108 \sqrt{5}-180\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(-108 \sqrt{5}-180\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{5}}{120}-\frac{3 n^{3}}{8}+\frac{3 n^{2}}{2}-\frac{62 n}{15}+4
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 182
a(7) = 492
a(n+2) = -a(n)+3*a(n+1)-1/120*n*(n^4+5*n^3-55*n^2-5*n-66), n >= 8
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) = 63
a \! \left(6\right) = 182
a \! \left(7\right) = 492
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n^{4}+5 n^{3}-55 n^{2}-5 n -66\right)}{120}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15675/
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[12,x]*F[4,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[20,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]
F[20,x] = F[21,x]+F[22,x]+F[30,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[12,x]*F[13,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[21,x]+F[33,x]+F[62,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[49,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[21,x]+F[26,x]+F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[21,x]+F[26,x]+F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[26,x]
F[44,x] = 2*F[21,x]+F[45,x]+F[48,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[25,x]+F[47,x]
F[47,x] = F[45,x]
F[48,x] = F[12,x]*F[42,x]
F[49,x] = F[32,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[61,x]
F[53,x] = F[21,x]+F[54,x]+F[59,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[10,x]+F[40,x]
F[57,x] = F[53,x]+F[58,x]
F[58,x] = F[51,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[14,x]+F[53,x]
F[61,x] = F[51,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[60,x]+F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[24,x]
F[68,x] = 2*F[21,x]+F[69,x]+F[75,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[40,x]+F[72,x]
F[72,x] = 2*F[21,x]+F[45,x]+F[73,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[25,x]+F[72,x]
F[75,x] = F[12,x]*F[64,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_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{20}\! \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_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)+F_{62}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{49}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{26}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{49}\! \left(x \right) = F_{32}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{53}\! \left(x \right) = F_{21}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{51}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{61}\! \left(x \right) = F_{51}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{60}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{24}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{69}\! \left(x \right)+F_{75}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{40}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{45}\! \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_{25}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \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_3(x))
Eq(F_3(x), F_12(x)*F_4(x))
Eq(F_4(x), F_31(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_12(x)*F_13(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_21(x) + F_33(x) + F_62(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_49(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_21(x) + F_26(x) + F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_21(x) + F_26(x) + F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_26(x))
Eq(F_44(x), 2*F_21(x) + F_45(x) + F_48(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_25(x) + F_47(x))
Eq(F_47(x), F_45(x))
Eq(F_48(x), F_12(x)*F_42(x))
Eq(F_49(x), F_32(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_61(x))
Eq(F_53(x), F_21(x) + F_54(x) + F_59(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_10(x) + F_40(x))
Eq(F_57(x), F_53(x) + F_58(x))
Eq(F_58(x), F_51(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_14(x) + F_53(x))
Eq(F_61(x), F_51(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_60(x) + F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_24(x))
Eq(F_68(x), 2*F_21(x) + F_69(x) + F_75(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_40(x) + F_72(x))
Eq(F_72(x), 2*F_21(x) + F_45(x) + F_73(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_25(x) + F_72(x))
Eq(F_75(x), F_12(x)*F_64(x))
Pack JSON:
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Specification JSON:
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