0123_0132_0231_2301
Counting sequence:
1, 1, 2, 6, 20, 64, 190, 524, 1356, 3330, 7842, 17870, 39700, 86508, 185782, 394680, 831772, 1742574, 3634714, 7556458, 15670148, 32431848, 67016174, 138297028, 285071212, 587029946, 1207750482, 2482776486, 5099973108, 10468625732, 21474414694, 44022919152, 90193733692, 184682919270, 377956340554, 773093211522, 1580546928484, 3229814219776, 6597068412446, 13469015900220, 27487788948620, 56075091043570, 114349207065346, 233096462589950, 474989020398804, 967570229312028, 1970324833482774, 4011018414241448, 8162774320295260, 16607023621149534, 33776997199994746, 68679894311569434, 139611588442063684, 283726776517281432, 576460752295675726, 1170935903107840564, 2377900603242337196, 4827858800531031978, 9799832789147142578, 19887895954456071382, 40352252661226554292, 81857426827071922292, 166020696663370552518, 336653079345182626272, 682529530727235356796, 1383505805528196868310, 2803905099203830801066, 5681597174702519215410, 11510768301994735787812, 23316684509168846978736, 47223664828696423919998, 95627921278110285292716, 193617025797655421289676, 391956418078180517952994, 793357569122100358675298, 1605604604175679332853806, 3248988140214315864501908, 6573534144154546092079180, 13298184015760920873365110, 26898599486425499085633944, 54401661882658312806859164, 110012249584931254839832014, 222442350809091768083817690, 449720404896642052924705994, 909112216350201139308989188, 1837567245814236345479070792, 3713820117856140824618589358, 7505011488167617916492478372, 15164765481245908367425910764, 30639015972313161803659837338, 61897001964269013744857362450, 125031943967823407764707092998, 252539768014217576079311031668, 510031296185576673258322753956, 1029966112685436388715948543270, 2079739265999438861830399223248, 4199092613256009892457692946620, 8477413389026284122509059021382, 17113283103081096920205342060042, 34543478856219251190785003271906, 69720783012552617082318509035236
Generating function in Maple syntax:
-(9*x^7-35*x^6+77*x^5-97*x^4+75*x^3-35*x^2+9*x-1)/(2*x-1)^2/(x-1)^6
Generating function in latex syntax:
-\frac{9 x^{7}-35 x^{6}+77 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-9*x**7 + 35*x**6 - 77*x**5 + 97*x**4 - 75*x**3 + 35*x**2 - 9*x + 1)/((x - 1)**6*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^6*F(x)+9*x^7-35*x^6+77*x^5-97*x^4+75*x^3-35*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+9 x^{7}-35 x^{6}+77 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
-7/12*n^3-n^2-22/5*n-4-1/60*n^5+2^(-1+n)*n+5*2^n
Explicit closed form in latex syntax:
-\frac{7 n^{3}}{12}-n^{2}-\frac{22 n}{5}-4-\frac{n^{5}}{60}+2^{-1+n} n +5 \,2^{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) = 190
a(7) = 524
a(n+2) = -1/60*n^5+1/6*n^4-7/12*n^3+11/6*n^2-4*a(n)+4*a(n+1)-7/5*n+2, 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) = 64
a \! \left(6\right) = 190
a \! \left(7\right) = 524
a \! \left(n +2\right) = -\frac{n^{5}}{60}+\frac{n^{4}}{6}-\frac{7 n^{3}}{12}+\frac{11 n^{2}}{6}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{7 n}{5}+2, \quad n \geq 8
Specification 1
Strategy pack name: insertion_row_and_col_placements
Tree: http://permpal.com/tree/21132/
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[15,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[31,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[8,x]
F[8,x] = F[10,x]+F[9,x]
F[9,x] = F[1,x]+F[7,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[10,x]+F[12,x]+F[16,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]
F[15,x] = x
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[15,x]*F[18,x]
F[18,x] = F[19,x]+F[26,x]
F[19,x] = F[12,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[15,x]*F[24,x]
F[24,x] = F[13,x]+F[21,x]
F[25,x] = F[15,x]*F[20,x]
F[26,x] = F[13,x]+F[27,x]
F[27,x] = F[22,x]+F[28,x]+F[30,x]
F[28,x] = F[15,x]*F[29,x]
F[29,x] = F[13,x]+F[27,x]
F[30,x] = F[15,x]*F[26,x]
F[31,x] = F[22,x]+F[32,x]+F[68,x]
F[32,x] = F[15,x]*F[33,x]
F[33,x] = F[34,x]+F[38,x]
F[34,x] = F[13,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[15,x]*F[37,x]
F[37,x] = F[13,x]+F[35,x]
F[38,x] = F[39,x]+F[47,x]
F[39,x] = F[40,x]
F[40,x] = F[15,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[13,x]+F[39,x]
F[44,x] = F[13,x]*F[45,x]
F[45,x] = F[12,x]+F[26,x]
F[46,x] = F[13,x]*F[20,x]
F[47,x] = F[48,x]
F[48,x] = F[15,x]*F[49,x]
F[49,x] = F[50,x]+F[56,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[13,x]*F[53,x]
F[53,x] = 2*F[54,x]
F[54,x] = F[12,x]*F[13,x]
F[55,x] = F[13,x]^2*F[12,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[13,x]*F[58,x]
F[58,x] = F[27,x]+F[59,x]
F[59,x] = F[13,x]*F[60,x]
F[60,x] = F[20,x]+F[27,x]
F[61,x] = F[47,x]+F[62,x]
F[62,x] = F[13,x]*F[63,x]
F[63,x] = 2*F[22,x]+F[64,x]+F[66,x]
F[64,x] = F[15,x]*F[65,x]
F[65,x] = F[35,x]+F[63,x]
F[66,x] = F[15,x]*F[67,x]
F[67,x] = F[27,x]+F[63,x]
F[68,x] = F[15,x]*F[69,x]
F[69,x] = F[70,x]+F[77,x]
F[70,x] = F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[12,x]*F[13,x]
F[73,x] = F[74,x]
F[74,x] = F[15,x]*F[75,x]
F[75,x] = F[73,x]+F[76,x]
F[76,x] = F[11,x]*F[15,x]
F[77,x] = F[31,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]*F[13,x]*F[15,x]*F[80,x]
F[80,x] = F[13,x]+F[20,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{31}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right) F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right) F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{32}\! \left(x \right)+F_{68}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{15}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{56}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{27}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{61}\! \left(x \right) = F_{47}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{13}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{35}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{15}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{27}\! \left(x \right)+F_{63}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{77}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{77}\! \left(x \right) = F_{31}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{13}\! \left(x \right)+F_{20}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_31(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_15(x)*F_8(x))
Eq(F_8(x), F_10(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_7(x))
Eq(F_10(x), F_11(x)*F_15(x))
Eq(F_11(x), F_10(x) + F_12(x) + F_16(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x))
Eq(F_15(x), x)
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_15(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_26(x))
Eq(F_19(x), F_12(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_15(x)*F_24(x))
Eq(F_24(x), F_13(x) + F_21(x))
Eq(F_25(x), F_15(x)*F_20(x))
Eq(F_26(x), F_13(x) + F_27(x))
Eq(F_27(x), F_22(x) + F_28(x) + F_30(x))
Eq(F_28(x), F_15(x)*F_29(x))
Eq(F_29(x), F_13(x) + F_27(x))
Eq(F_30(x), F_15(x)*F_26(x))
Eq(F_31(x), F_22(x) + F_32(x) + F_68(x))
Eq(F_32(x), F_15(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_38(x))
Eq(F_34(x), F_13(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_15(x)*F_37(x))
Eq(F_37(x), F_13(x) + F_35(x))
Eq(F_38(x), F_39(x) + F_47(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_15(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_13(x) + F_39(x))
Eq(F_44(x), F_13(x)*F_45(x))
Eq(F_45(x), F_12(x) + F_26(x))
Eq(F_46(x), F_13(x)*F_20(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_15(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_56(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_13(x)*F_53(x))
Eq(F_53(x), 2*F_54(x))
Eq(F_54(x), F_12(x)*F_13(x))
Eq(F_55(x), F_12(x)*F_13(x)**2)
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_13(x)*F_58(x))
Eq(F_58(x), F_27(x) + F_59(x))
Eq(F_59(x), F_13(x)*F_60(x))
Eq(F_60(x), F_20(x) + F_27(x))
Eq(F_61(x), F_47(x) + F_62(x))
Eq(F_62(x), F_13(x)*F_63(x))
Eq(F_63(x), 2*F_22(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_15(x)*F_65(x))
Eq(F_65(x), F_35(x) + F_63(x))
Eq(F_66(x), F_15(x)*F_67(x))
Eq(F_67(x), F_27(x) + F_63(x))
Eq(F_68(x), F_15(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_77(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_12(x)*F_13(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_15(x)*F_75(x))
Eq(F_75(x), F_73(x) + F_76(x))
Eq(F_76(x), F_11(x)*F_15(x))
Eq(F_77(x), F_31(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x)*F_13(x)*F_15(x)*F_80(x))
Eq(F_80(x), F_13(x) + F_20(x))
Pack JSON:
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Specification JSON:
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