0132_0231_1203_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 139, 334, 767, 1710, 3739, 8070, 17263, 36694, 77627, 163614, 343807, 720606, 1507003, 3145366, 6553199, 13631046, 28311067, 58719726, 121634239, 251657614, 520093019, 1073741094, 2214591727, 4563401910, 9395240059, 19327351870, 39728446463, 81604377534, 167503723387, 343597382454, 704374635247, 1443109010086, 2954937498203, 6047313951246, 12369505810879, 25288767437166, 51677046503707, 105553116264646, 215504279042159, 439804651108374, 897201488263099, 1829587348617054, 3729543441413887, 7599824371185310, 15481123719083579, 31525197391590870, 64176294690026863, 130604389193741574, 265712378014856347, 540431955284456494, 1098878309078397887, 2233785415175762766, 4539628424389456603, 9223372036854772326, 18734974449861259759, 38046409652025946486, 77245740808658743547, 156797324626531184766, 318206335271489761279, 645636042579834302334, 1309718829233378160379, 2656331146614175428214, 5386449269523189067247, 10920472491636054551910, 22136092888451461934299, 44862481587261629525070, 90905554795240670358463, 184172292831916163328814, 373066952146701971876507, 755578637259143234185734, 1530046740449765049231727, 3097872412762487260178646, 6271302689250888843782203, 12693721105953606334408606, 25689673666810869962499839, 51983810243429054512359006, 105176546306472738199430587, 212770944252174734748280086, 430377591782807986195391599, 870426590122533005788439494, 1760195993358900078372184859, 3559077612945468290334974574, 7195526478346272847851151807, 14545795461603218230064701710, 29401075933027781528854192219, 59421121885698253195157954470, 120080183810681886665215041263, 242636247699934533880228339254, 490224255557010588860053183867, 990352031428304219919299370174, 2000511103485174524236984736767, 4040636288227481217270741457726, 8160500738969226772135026875003, 16479457802966982219457141660086, 33275828255991021789288459131119
Generating function in Maple syntax:
(3*x^5-6*x^4+14*x^3-14*x^2+6*x-1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{3 x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(3*x**5 - 6*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)-3*x^5+6*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)-3 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/4*(n+5)*2^n-n^2-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(n +5\right) 2^{n}}{4}-n^{2}-1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(n+2) = -n^2-4*a(n)+4*a(n+1)+4*n-1, 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) = 19
a \! \left(5\right) = 54
a \! \left(n +2\right) = -n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+4 n -1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19012/
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[22,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[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[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[40,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[34,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[9,x]
F[30,x] = F[14,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[31,x]
F[34,x] = F[23,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[27,x]
F[39,x] = F[35,x]
F[40,x] = F[18,x]+F[41,x]+F[70,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[50,x]
F[43,x] = F[44,x]+F[6,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[14,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[47,x]
F[50,x] = F[40,x]+F[51,x]
F[51,x] = 2*F[18,x]+F[52,x]+F[56,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[44,x]
F[55,x] = F[51,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]+F[64,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[14,x]
F[63,x] = F[59,x]
F[64,x] = 2*F[18,x]+F[65,x]+F[69,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[48,x]
F[68,x] = F[64,x]
F[69,x] = F[12,x]*F[58,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]+F[85,x]
F[72,x] = F[23,x]+F[73,x]
F[73,x] = F[18,x]+F[74,x]+F[84,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[78,x]
F[76,x] = F[10,x]+F[77,x]
F[77,x] = F[49,x]
F[78,x] = F[73,x]+F[79,x]
F[79,x] = 2*F[18,x]+F[69,x]+F[80,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[77,x]
F[83,x] = F[79,x]
F[84,x] = F[12,x]*F[72,x]
F[85,x] = F[59,x]+F[86,x]
F[86,x] = 2*F[18,x]+F[65,x]+F[87,x]
F[87,x] = F[12,x]*F[85,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_{22}\! \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_{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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{40}\! \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_{34}\! \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_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{23}\! \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_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{27}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{70}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{6}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{40}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{44}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \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_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{64}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{14}\! \left(x \right)
F_{63}\! \left(x \right) = F_{59}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{48}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{85}\! \left(x \right)
F_{72}\! \left(x \right) = F_{23}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{74}\! \left(x \right)+F_{84}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{76}\! \left(x \right) = F_{10}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{49}\! \left(x \right)
F_{78}\! \left(x \right) = F_{73}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{69}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{77}\! \left(x \right)
F_{83}\! \left(x \right) = F_{79}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{85}\! \left(x \right) = F_{59}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \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_22(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_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_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_40(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_34(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_29(x))
Eq(F_29(x), F_30(x) + F_9(x))
Eq(F_30(x), F_14(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_31(x))
Eq(F_34(x), F_23(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_38(x) + F_39(x))
Eq(F_38(x), F_27(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_18(x) + F_41(x) + F_70(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_50(x))
Eq(F_43(x), F_44(x) + F_6(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_14(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_47(x))
Eq(F_50(x), F_40(x) + F_51(x))
Eq(F_51(x), 2*F_18(x) + F_52(x) + F_56(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_44(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x) + F_64(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_14(x))
Eq(F_63(x), F_59(x))
Eq(F_64(x), 2*F_18(x) + F_65(x) + F_69(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_48(x))
Eq(F_68(x), F_64(x))
Eq(F_69(x), F_12(x)*F_58(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_85(x))
Eq(F_72(x), F_23(x) + F_73(x))
Eq(F_73(x), F_18(x) + F_74(x) + F_84(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_78(x))
Eq(F_76(x), F_10(x) + F_77(x))
Eq(F_77(x), F_49(x))
Eq(F_78(x), F_73(x) + F_79(x))
Eq(F_79(x), 2*F_18(x) + F_69(x) + F_80(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_77(x))
Eq(F_83(x), F_79(x))
Eq(F_84(x), F_12(x)*F_72(x))
Eq(F_85(x), F_59(x) + F_86(x))
Eq(F_86(x), 2*F_18(x) + F_65(x) + F_87(x))
Eq(F_87(x), F_12(x)*F_85(x))
Pack JSON:
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Specification JSON:
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