0132_0213_0231_2301_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 117, 241, 464, 860, 1568, 2856, 5249, 9785, 18527, 35587, 69174, 135690, 267922, 531426, 1057295, 2107695, 4206937, 8403621, 16794924, 33575176, 67133012, 134245676, 268467629, 536907765, 1073783843, 2147531351, 4295021234, 8589995350, 17179937382, 34359814662, 68719561819, 137439048075, 274878011837, 549755929881, 1099511755720, 2199023396340, 4398046665672, 8796093191536, 17592186229529, 35184372290801, 70368744397607, 140737488594411, 281474976970094, 562949953702370, 1125899907146618, 2251799814013546, 4503599627724519, 9007199255122215, 18014398509891937, 36028797019404237, 72057594038400164, 144115188076361760, 288230376152253052, 576460752304002036, 1152921504607464645, 2305843009214352685, 4611686018428089707, 9223372036855522751, 18446744073710345834, 36893488147419946926, 73786976294839101902, 147573952589677362446, 295147905179353831859, 590295810358706716675, 1180591620717412429893, 2361183241434823797441, 4722366482869646471104, 9444732965739291754380, 18889465931478582254192, 37778931862957163184312, 75557863725914324972209, 151115727451828648472745, 302231454903657295395567, 604462909807314589159891, 1208925819614629176604070, 2417851639229258351404730, 4835703278458516700915042, 9671406556917033399841266, 19342813113834066797595839, 38685626227668133593003551, 77371252455336267183713897, 154742504910672534365025781, 309485009821345068727536924, 618970019642690137452442680, 1237940039285380274902133668, 2475880078570760549801391036, 4951760157141521099599776989, 9903520314283042199196415845, 19807040628566084398389556147, 39614081257132168796775694887, 79228162514264337593547825954, 158456325028528675187091937030, 316912650057057350374180003382, 633825300114114700748355975446, 1267650600228229401496707753995
Generating function in Maple syntax:
(2*x^7+x^6-8*x^5+12*x^4-18*x^3+15*x^2-6*x+1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{7}+x^{6}-8 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**7 + x**6 - 8*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x-1)^5*F(x)+2*x^7+x^6-8*x^5+12*x^4-18*x^3+15*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}+x^{6}-8 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-6+1/24*n^4+5/12*n^3-85/24*n^2+85/12*n+2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -6+\frac{n^{4}}{24}+\frac{5 n^{3}}{12}-\frac{85 n^{2}}{24}+\frac{85 n}{12}+2^{n} & \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) = 51
a(6) = 117
a(7) = 241
a(n+1) = -1/24*n^4-1/4*n^3+121/24*n^2+2*a(n)-51/4*n+10, 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) = 19
a \! \left(5\right) = 51
a \! \left(6\right) = 117
a \! \left(7\right) = 241
a \! \left(n +1\right) = -\frac{n^{4}}{24}-\frac{n^{3}}{4}+\frac{121 n^{2}}{24}+2 a \! \left(n \right)-\frac{51 n}{4}+10, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18215/
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[14,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[7,x]
F[14,x] = F[15,x]+F[30,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[6,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[28,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[27,x]
F[27,x] = F[22,x]*F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[20,x]
F[30,x] = F[31,x]+F[32,x]+F[74,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[48,x]
F[34,x] = F[35,x]+F[7,x]
F[35,x] = F[31,x]+F[36,x]+F[41,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[23,x]
F[44,x] = F[45,x]
F[45,x] = F[31,x]+F[41,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[11,x]
F[48,x] = F[30,x]+F[49,x]
F[49,x] = 2*F[31,x]+F[50,x]+F[60,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]+F[58,x]
F[53,x] = F[31,x]+F[54,x]+F[56,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[11,x]+F[53,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[23,x]+F[53,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[52,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[66,x]
F[62,x] = F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[23,x]+F[63,x]
F[66,x] = F[67,x]
F[67,x] = 2*F[31,x]+F[68,x]+F[70,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[26,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[63,x]
F[73,x] = F[67,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[57,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[31,x]+F[78,x]+F[80,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[29,x]+F[7,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[22,x]+F[82,x]
F[82,x] = F[77,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_{14}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
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_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \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_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \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_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{74}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{48}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{41}\! \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_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{23}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)+F_{41}\! \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_{11}\! \left(x \right)
F_{48}\! \left(x \right) = F_{30}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{50}\! \left(x \right)+F_{60}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{53}\! \left(x \right) = F_{31}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{23}\! \left(x \right)+F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \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_{66}\! \left(x \right)
F_{62}\! \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_{23}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{26}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \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_{63}\! \left(x \right)
F_{73}\! \left(x \right) = F_{67}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{57}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{31}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{22}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{77}\! \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_14(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_7(x))
Eq(F_14(x), F_15(x) + F_30(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_6(x))
Eq(F_18(x), F_15(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_22(x) + F_28(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_22(x)*F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_20(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_74(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_48(x))
Eq(F_34(x), F_35(x) + F_7(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_41(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_23(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_31(x) + F_41(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_11(x))
Eq(F_48(x), F_30(x) + F_49(x))
Eq(F_49(x), 2*F_31(x) + F_50(x) + F_60(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x) + F_58(x))
Eq(F_53(x), F_31(x) + F_54(x) + F_56(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_11(x) + F_53(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_23(x) + F_53(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_52(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_66(x))
Eq(F_62(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_23(x) + F_63(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), 2*F_31(x) + F_68(x) + F_70(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_26(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_63(x))
Eq(F_73(x), F_67(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_57(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_31(x) + F_78(x) + F_80(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_29(x) + F_7(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_22(x) + F_82(x))
Eq(F_82(x), F_77(x))
Pack JSON:
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Specification JSON:
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