0132_0213_1230_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 53, 132, 308, 693, 1527, 3322, 7166, 15363, 32777, 69648, 147480, 311329, 655403, 1376310, 2883650, 6029391, 12583005, 26214508, 54526076, 113246349, 234881183, 486539442, 1006633158, 2080375003, 4294967537, 8858370312, 18253611296, 37580964153, 77309411667, 158913790318, 326417514890, 670014898599, 1374389535173, 2817498546660, 5772436046340, 11819749999141, 24189255811655, 49478023250538, 101155069756046, 206708186022579, 422212465066713, 862017116177152, 1759218604442408, 3588805953061713, 7318349394477947, 14918173765665702, 30399297484751826, 61924494876345343, 126100789566374957, 256705178760119388, 522417556774978700, 1062849512059438269, 2161727821137839343, 4395513236313605410, 8935141660703065430, 18158513697557841291, 36893488147419104705, 74939897799445054968, 152185638608103802416, 308982963234634991209, 627189298506124756643, 1272825341085959063262, 2582544170319337228058, 5238875316933512660823, 10625324586456701732757, 21545797078092756289492, 43681889966544218228756, 88544371553805847758933, 179449926349046518122647, 363622219180962681456858, 736689171327664653338910, 1492267808586807887530339, 3022314549036572936767913, 6120186961799060196952560, 12391489651049949040740920, 25085210757003555375155841, 50774884423814425337662155, 102758694667243479850027798, 207935240973716218049465186, 420706185225890952797752239, 851083777008698938993150973, 1721510367131231944781597772, 3481706360490132023153790108, 7040783973435600313488772333, 14236310451781873161339931967, 28782105913385091391404641682, 58183181846412872920258842086, 117604303732111126115416804923, 237684487542793012780631854737, 480320735242727546660860202728, 970544990799738135520913395520, 1960897022228042355440212774809, 3961408125713216879677197520883, 8002044413940698096947938988110, 16162545152909924869082965872810, 32642002955876907088540107542791
Generating function in Maple syntax:
-(4*x^7-4*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{4 x^{7}-4 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:
(-4*x**7 + 4*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)+4*x^7-4*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)+4 x^{7}-4 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/4*(n+3)*2^n+1/2*n^2-13/2*n+9)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(n +3\right) 2^{n}}{4}+\frac{n^{2}}{2}-\frac{13 n}{2}+9 & \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) = 53
a(6) = 132
a(7) = 308
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-17/2*n+22, 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) = 53
a \! \left(6\right) = 132
a \! \left(7\right) = 308
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{17 n}{2}+22, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18503/
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[21,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[19,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[22,x]+F[34,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[25,x]
F[28,x] = F[22,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[26,x]
F[33,x] = F[29,x]
F[34,x] = F[35,x]+F[36,x]+F[68,x]
F[35,x] = 0
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[53,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[35,x]+F[40,x]+F[45,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[26,x]+F[42,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[44,x]+F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[12,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[47,x]
F[53,x] = F[34,x]+F[54,x]
F[54,x] = 2*F[35,x]+F[55,x]+F[59,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[39,x]
F[58,x] = F[54,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[29,x]+F[62,x]
F[62,x] = 2*F[35,x]+F[63,x]+F[67,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[42,x]
F[66,x] = F[62,x]
F[67,x] = F[12,x]*F[61,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[83,x]
F[70,x] = F[22,x]+F[71,x]
F[71,x] = F[35,x]+F[72,x]+F[82,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[74,x]+F[76,x]
F[74,x] = F[10,x]+F[75,x]
F[75,x] = F[35,x]+F[40,x]+F[43,x]
F[76,x] = F[71,x]+F[77,x]
F[77,x] = 2*F[35,x]+F[67,x]+F[78,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[80,x]+F[81,x]
F[80,x] = F[75,x]
F[81,x] = F[77,x]
F[82,x] = F[12,x]*F[70,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[30,x]
F[85,x] = 2*F[35,x]+F[63,x]+F[86,x]
F[86,x] = F[12,x]*F[83,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_{21}\! \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_{19}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{68}\! \left(x \right)
F_{35}\! \left(x \right) = 0
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_{53}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{26}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{34}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{39}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{29}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{42}\! \left(x \right)
F_{66}\! \left(x \right) = F_{62}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{83}\! \left(x \right)
F_{70}\! \left(x \right) = F_{22}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{35}\! \left(x \right)+F_{72}\! \left(x \right)+F_{82}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{10}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{76}\! \left(x \right) = F_{71}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{67}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{81}\! \left(x \right)
F_{80}\! \left(x \right) = F_{75}\! \left(x \right)
F_{81}\! \left(x \right) = F_{77}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{30}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{63}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \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_21(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_19(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_22(x) + F_34(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_25(x))
Eq(F_28(x), F_22(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_26(x))
Eq(F_33(x), F_29(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_68(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_53(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_45(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_26(x) + F_42(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_44(x) + F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_12(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_47(x))
Eq(F_53(x), F_34(x) + F_54(x))
Eq(F_54(x), 2*F_35(x) + F_55(x) + F_59(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_39(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_29(x) + F_62(x))
Eq(F_62(x), 2*F_35(x) + F_63(x) + F_67(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_42(x))
Eq(F_66(x), F_62(x))
Eq(F_67(x), F_12(x)*F_61(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_83(x))
Eq(F_70(x), F_22(x) + F_71(x))
Eq(F_71(x), F_35(x) + F_72(x) + F_82(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_76(x))
Eq(F_74(x), F_10(x) + F_75(x))
Eq(F_75(x), F_35(x) + F_40(x) + F_43(x))
Eq(F_76(x), F_71(x) + F_77(x))
Eq(F_77(x), 2*F_35(x) + F_67(x) + F_78(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_81(x))
Eq(F_80(x), F_75(x))
Eq(F_81(x), F_77(x))
Eq(F_82(x), F_12(x)*F_70(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_30(x))
Eq(F_85(x), 2*F_35(x) + F_63(x) + F_86(x))
Eq(F_86(x), F_12(x)*F_83(x))
Pack JSON:
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Specification JSON:
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