0123_1032_1230_2301
Counting sequence:
1, 1, 2, 6, 20, 58, 140, 292, 546, 940, 1518, 2330, 3432, 4886, 6760, 9128, 12070, 15672, 20026, 25230, 31388, 38610, 47012, 56716, 67850, 80548, 94950, 111202, 129456, 149870, 172608, 197840, 225742, 256496, 290290, 327318, 367780, 411882, 459836, 511860, 568178, 629020, 694622, 765226, 841080, 922438, 1009560, 1102712, 1202166, 1308200, 1421098, 1541150, 1668652, 1803906, 1947220, 2098908, 2259290, 2428692, 2607446, 2795890, 2994368, 3203230, 3422832, 3653536, 3895710, 4149728, 4415970, 4694822, 4986676, 5291930, 5610988, 5944260, 6292162, 6655116, 7033550, 7427898, 7838600, 8266102, 8710856, 9173320, 9653958, 10153240, 10671642, 11209646, 11767740, 12346418, 12946180, 13567532, 14210986, 14877060, 15566278, 16279170, 17016272, 17778126, 18565280, 19378288, 20217710, 21084112, 21978066, 22900150, 23850948
Generating function in Maple syntax:
(x^6-2*x^5-5*x^4+4*x^3-7*x^2+4*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{x^{6}-2 x^{5}-5 x^{4}+4 x^{3}-7 x^{2}+4 x -1}{\left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**6 - 2*x**5 - 5*x**4 + 4*x**3 - 7*x**2 + 4*x - 1)/(x - 1)**5
Implicit equation for the generating function in Maple syntax:
-(x-1)^5*F(x)+x^6-2*x^5-5*x^4+4*x^3-7*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{5} F \! \left(x \right)+x^{6}-2 x^{5}-5 x^{4}+4 x^{3}-7 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/12*(3*n^2-14*n+24)*(n^2-1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(3 n^{2}-14 n +24\right) \left(n^{2}-1\right)}{12} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 58
a(6) = 140
a(n) = 1/12*(-1+n)*(n+1)*(3*n^2-14*n+24), 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) = 20
a \! \left(5\right) = 58
a \! \left(6\right) = 140
a \! \left(n \right) = \frac{\left(-1+n \right) \left(n +1\right) \left(3 n^{2}-14 n +24\right)}{12}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15149/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[34,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]+F[16,x]+F[28,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[11,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[15,x]+F[16,x]+F[20,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[11,x]+F[25,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[15,x]+F[28,x]+F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[4,x]
F[33,x] = F[15,x]+F[20,x]+F[31,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[15,x]+F[36,x]+F[79,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[48,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[1,x]+F[4,x]
F[42,x] = F[15,x]+F[16,x]+F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[11,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[4,x]+F[45,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[15,x]+F[36,x]+F[43,x]
F[50,x] = F[15,x]+F[51,x]+F[68,x]+F[77,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[60,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[4,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[45,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = 2*F[15,x]+F[51,x]+F[58,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[45,x]+F[65,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[45,x]
F[71,x] = F[72,x]
F[72,x] = 2*F[15,x]+F[68,x]+F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[55,x]
F[76,x] = 2*F[15,x]+F[58,x]+F[73,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[25,x]+F[65,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[81,x]+F[83,x]
F[81,x] = F[11,x]+F[82,x]
F[82,x] = F[46,x]
F[83,x] = F[84,x]+F[88,x]
F[84,x] = F[15,x]+F[79,x]+F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[39,x]+F[87,x]
F[87,x] = F[15,x]+F[43,x]+F[85,x]
F[88,x] = 2*F[15,x]+F[68,x]+F[73,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{34}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{28}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{79}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{43}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{68}\! \left(x \right)+F_{77}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{60}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{45}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{58}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{45}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{45}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{68}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{55}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{73}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{25}\! \left(x \right)+F_{65}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{81}\! \left(x \right) = F_{11}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{46}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{88}\! \left(x \right)
F_{84}\! \left(x \right) = F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{39}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{85}\! \left(x \right)
F_{88}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{68}\! \left(x \right)+F_{73}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_34(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_28(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_11(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_11(x) + F_25(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_15(x) + F_28(x) + F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_4(x))
Eq(F_33(x), F_15(x) + F_20(x) + F_31(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_15(x) + F_36(x) + F_79(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_48(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_1(x) + F_4(x))
Eq(F_42(x), F_15(x) + F_16(x) + F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_11(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_4(x) + F_45(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_15(x) + F_36(x) + F_43(x))
Eq(F_50(x), F_15(x) + F_51(x) + F_68(x) + F_77(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_60(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_4(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_45(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), 2*F_15(x) + F_51(x) + F_58(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_45(x) + F_65(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_45(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), 2*F_15(x) + F_68(x) + F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_55(x))
Eq(F_76(x), 2*F_15(x) + F_58(x) + F_73(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_25(x) + F_65(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_83(x))
Eq(F_81(x), F_11(x) + F_82(x))
Eq(F_82(x), F_46(x))
Eq(F_83(x), F_84(x) + F_88(x))
Eq(F_84(x), F_15(x) + F_79(x) + F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_39(x) + F_87(x))
Eq(F_87(x), F_15(x) + F_43(x) + F_85(x))
Eq(F_88(x), 2*F_15(x) + F_68(x) + F_73(x))
Pack JSON:
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Specification JSON:
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