0123_1230_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 64, 185, 477, 1107, 2347, 4612, 8504, 14862, 24818, 39859, 61895, 93333, 137157, 197014, 277306, 383288, 521172, 698237, 922945, 1205063, 1555791, 1987896, 2515852, 3155986, 3926630, 4848279, 5943755, 7238377, 8760137, 10539882, 12611502, 15012124, 17782312, 20966273, 24612069, 28771835, 33502003, 38863532, 44922144, 51748566, 59418778, 68014267, 77622287, 88336125, 100255373, 113486206, 128141666, 144341952, 162214716, 181895365, 203527369, 227262575, 253261527, 281693792, 312738292, 346583642, 383428494, 423481887, 466963603, 514104529, 565147025, 620345298, 679965782, 744287524, 813602576, 888216393, 968448237, 1054631587, 1147114555, 1246260308, 1352447496, 1466070686, 1587540802, 1717285571, 1855749975, 2003396709, 2160706645, 2328179302, 2506333322, 2695706952, 2896858532, 3110366989, 3336832337, 3576876183, 3831142239, 4100296840, 4385029468, 4686053282, 5004105654, 5339948711, 5694369883, 6068182457, 6462226137, 6877367610, 7314501118, 7774549036
Generating function in Maple syntax:
-(3*x^6-6*x^5+20*x^4-22*x^3+16*x^2-6*x+1)/(x-1)^7
Generating function in latex syntax:
-\frac{3 x^{6}-6 x^{5}+20 x^{4}-22 x^{3}+16 x^{2}-6 x +1}{\left(x -1\right)^{7}}
Generating function in sympy syntax:
(-3*x**6 + 6*x**5 - 20*x**4 + 22*x**3 - 16*x**2 + 6*x - 1)/(x - 1)**7
Implicit equation for the generating function in Maple syntax:
(x-1)^7*F(x)+3*x^6-6*x^5+20*x^4-22*x^3+16*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{7} F \! \left(x \right)+3 x^{6}-6 x^{5}+20 x^{4}-22 x^{3}+16 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
1+1/120*n^6-7/120*n^5+1/4*n^4-11/24*n^3+89/120*n^2-29/60*n
Explicit closed form in latex syntax:
1+\frac{1}{120} n^{6}-\frac{7}{120} n^{5}+\frac{1}{4} n^{4}-\frac{11}{24} n^{3}+\frac{89}{120} n^{2}-\frac{29}{60} n
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 185
a(n) = 1+1/120*n^6-7/120*n^5+1/4*n^4-11/24*n^3+89/120*n^2-29/60*n, 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) = 64
a \! \left(6\right) = 185
a \! \left(n \right) = 1+\frac{1}{120} n^{6}-\frac{7}{120} n^{5}+\frac{1}{4} n^{4}-\frac{11}{24} n^{3}+\frac{89}{120} n^{2}-\frac{29}{60} n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15185/
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[33,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[22,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[11,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[19,x]
F[27,x] = F[23,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[11,x]+F[30,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[15,x]+F[35,x]+F[88,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[47,x]
F[37,x] = F[38,x]+F[7,x]
F[38,x] = F[15,x]+F[16,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[29,x]+F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[31,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[19,x]+F[46,x]
F[46,x] = F[44,x]
F[47,x] = F[34,x]+F[48,x]
F[48,x] = F[15,x]+F[49,x]+F[81,x]+F[87,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[58,x]
F[51,x] = F[38,x]+F[52,x]
F[52,x] = F[15,x]+F[24,x]+F[53,x]+F[57,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[30,x]+F[46,x]
F[56,x] = F[43,x]
F[57,x] = 0
F[58,x] = F[48,x]+F[59,x]
F[59,x] = F[15,x]+F[60,x]+F[73,x]+F[79,x]+F[80,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[67,x]
F[62,x] = F[52,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[46,x]
F[67,x] = F[59,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[63,x]
F[72,x] = F[68,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[66,x]+F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[23,x]+F[76,x]
F[79,x] = 0
F[80,x] = 0
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[55,x]+F[83,x]
F[83,x] = F[76,x]+F[84,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[14,x]+F[84,x]
F[87,x] = 0
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[29,x]+F[90,x]
F[90,x] = F[84,x]+F[91,x]
F[91,x] = F[92,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[7,x]+F[91,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_{33}\! \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_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \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_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \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_{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_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{88}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{47}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{31}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{34}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{81}\! \left(x \right)+F_{87}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{58}\! \left(x \right)
F_{51}\! \left(x \right) = F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{30}\! \left(x \right)+F_{46}\! \left(x \right)
F_{56}\! \left(x \right) = F_{43}\! \left(x \right)
F_{57}\! \left(x \right) = 0
F_{58}\! \left(x \right) = F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{73}\! \left(x \right)+F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{67}\! \left(x \right)
F_{62}\! \left(x \right) = F_{52}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{46}\! \left(x \right)
F_{67}\! \left(x \right) = F_{59}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{63}\! \left(x \right)
F_{72}\! \left(x \right) = F_{68}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{66}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{23}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = 0
F_{80}\! \left(x \right) = 0
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{55}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{76}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \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_{14}\! \left(x \right)+F_{84}\! \left(x \right)
F_{87}\! \left(x \right) = 0
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{29}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{84}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{7}\! \left(x \right)+F_{91}\! \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_33(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_22(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_11(x))
Eq(F_22(x), F_14(x) + F_23(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) + F_27(x))
Eq(F_26(x), F_19(x))
Eq(F_27(x), F_23(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_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_11(x) + F_30(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_15(x) + F_35(x) + F_88(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_47(x))
Eq(F_37(x), F_38(x) + F_7(x))
Eq(F_38(x), F_15(x) + F_16(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_29(x) + F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_31(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_19(x) + F_46(x))
Eq(F_46(x), F_44(x))
Eq(F_47(x), F_34(x) + F_48(x))
Eq(F_48(x), F_15(x) + F_49(x) + F_81(x) + F_87(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_58(x))
Eq(F_51(x), F_38(x) + F_52(x))
Eq(F_52(x), F_15(x) + F_24(x) + F_53(x) + F_57(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_30(x) + F_46(x))
Eq(F_56(x), F_43(x))
Eq(F_57(x), 0)
Eq(F_58(x), F_48(x) + F_59(x))
Eq(F_59(x), F_15(x) + F_60(x) + F_73(x) + F_79(x) + F_80(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_67(x))
Eq(F_62(x), F_52(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_46(x))
Eq(F_67(x), F_59(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_63(x))
Eq(F_72(x), F_68(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_66(x) + F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_23(x) + F_76(x))
Eq(F_79(x), 0)
Eq(F_80(x), 0)
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_55(x) + F_83(x))
Eq(F_83(x), F_76(x) + F_84(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_14(x) + F_84(x))
Eq(F_87(x), 0)
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_29(x) + F_90(x))
Eq(F_90(x), F_84(x) + F_91(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_7(x) + F_91(x))
Pack JSON:
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Specification JSON:
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