0132_0231_1302_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 138, 327, 738, 1617, 3483, 7432, 15777, 33392, 70528, 148701, 312980, 657587, 1379145, 2887270, 6033947, 12588666, 26221462, 54534531, 113256534, 234893349, 486553863, 1006650132, 2080394853, 4294990612, 8858396988, 18253641977, 37580999272, 77309451687, 158913835733, 326417566226, 670014956415, 1374389600062, 2817498619250, 5772436127295, 11819750089162, 24189255911481, 49478023360947, 101155069877856, 206708186156649, 422212465213944, 862017116338488, 1759218604618837, 3588805953254268, 7318349394687707, 14918173765893793, 30399297484999422, 61924494876613667, 126100789566665282, 256705178760433038, 522417556775317051, 1062849512059802750, 2161727821138231437, 4395513236314026655, 8935141660703517420, 18158513697558325677, 36893488147419623196, 74939897799445609332, 152185638608104394481, 308982963234635622864, 627189298506125429839, 1272825341085959780013, 2582544170319337990442, 5238875316933513470983, 10625324586456702592902, 21545797078092757201898, 43681889966544219195767, 88544371553805848782962, 179449926349046519206177, 363622219180962682602443, 736689171327664654549176, 1492267808586807888807985, 3022314549036572938115712, 6120186961799060198373360, 12391489651049949042237645, 25085210757003555376731492, 50774884423814425339319811, 102758694667243479851770617, 207935240973716218051296406, 420706185225890952799675179, 851083777008698938995169034, 1721510367131231944783714438, 3481706360490132023156008947, 7040783973435600313491096998, 14236310451781873161342366197, 28782105913385091391407189303, 58183181846412872920261507012, 117604303732111126115419591157, 237684487542793012780634766372, 480320735242727546660863243948, 970544990799738135520916570601, 1960897022228042355440216088120, 3961408125713216879677200976887, 8002044413940698096947942591365, 16162545152909924869082969627970, 32642002955876907088540111454607
Generating function in Maple syntax:
(x^7-11*x^6+29*x^5-48*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{x^{7}-11 x^{6}+29 x^{5}-48 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**7 - 11*x**6 + 29*x**5 - 48*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)-x^7+11*x^6-29*x^5+48*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)-x^{7}+11 x^{6}-29 x^{5}+48 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/4*2^n*n+3/4*2^n-1/24*n^2-3/4*n-1/4*n^3+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{2^{n} n}{4}+\frac{3 \,2^{n}}{4}-\frac{n^{2}}{24}-\frac{3 n}{4}-\frac{n^{3}}{4}+\frac{n^{4}}{24} & \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) = 54
a(6) = 138
a(7) = 327
a(n+2) = 1/24*n^4-7/12*n^3+35/24*n^2-4*a(n)+4*a(n+1)+1/12*n+1, 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) = 54
a \! \left(6\right) = 138
a \! \left(7\right) = 327
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{7 n^{3}}{12}+\frac{35 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{n}{12}+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21733/
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[21,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[21,x]*F[7,x]
F[7,x] = F[75,x]+F[8,x]
F[8,x] = F[54,x]+F[9,x]
F[9,x] = F[10,x]+F[45,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[21,x]
F[13,x] = F[14,x]+F[31,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[21,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[21,x]
F[21,x] = x
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[21,x]*F[25,x]
F[25,x] = F[1,x]+F[24,x]
F[26,x] = F[27,x]+F[28,x]+F[30,x]
F[27,x] = 0
F[28,x] = F[21,x]*F[29,x]
F[29,x] = F[19,x]+F[26,x]
F[30,x] = F[21,x]*F[22,x]
F[31,x] = F[23,x]+F[32,x]
F[32,x] = F[27,x]+F[33,x]+F[35,x]
F[33,x] = F[21,x]*F[34,x]
F[34,x] = F[15,x]+F[32,x]
F[35,x] = F[21,x]*F[36,x]
F[36,x] = F[22,x]+F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[39,x]
F[39,x] = F[21,x]*F[40,x]
F[40,x] = F[23,x]+F[38,x]
F[41,x] = 2*F[27,x]+F[42,x]+F[44,x]
F[42,x] = F[21,x]*F[43,x]
F[43,x] = F[26,x]+F[41,x]
F[44,x] = F[21,x]*F[37,x]
F[45,x] = F[11,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[21,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[15,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[21,x]*F[53,x]
F[53,x] = F[15,x]+F[51,x]
F[54,x] = F[55,x]+F[74,x]
F[55,x] = F[23,x]+F[56,x]
F[56,x] = F[27,x]+F[57,x]+F[70,x]
F[57,x] = F[21,x]*F[58,x]
F[58,x] = F[15,x]+F[59,x]
F[59,x] = F[27,x]+F[57,x]+F[60,x]
F[60,x] = F[21,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[23,x]
F[63,x] = F[64,x]
F[64,x] = F[27,x]+F[65,x]+F[68,x]
F[65,x] = F[21,x]*F[66,x]
F[66,x] = F[23,x]+F[67,x]
F[67,x] = F[65,x]
F[68,x] = F[21,x]*F[69,x]
F[69,x] = F[23,x]+F[64,x]
F[70,x] = F[21,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[23,x]
F[73,x] = F[64,x]
F[74,x] = F[52,x]
F[75,x] = F[76,x]
F[76,x] = F[25,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[23,x]*F[25,x]
F[79,x] = F[80,x]
F[80,x] = F[21,x]*F[81,x]
F[81,x] = F[69,x]+F[82,x]
F[82,x] = F[83,x]
F[83,x] = F[21,x]*F[25,x]*F[81,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_{21}\! \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_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{75}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{54}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \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_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{21}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = x
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_{21}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{21}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right) F_{22}\! \left(x \right)
F_{31}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{21}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{21}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{23}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{21}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{26}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{21}\! \left(x \right) F_{37}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right) F_{48}\! \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_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{74}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{27}\! \left(x \right)+F_{57}\! \left(x \right)+F_{70}\! \left(x \right)
F_{57}\! \left(x \right) = F_{21}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{27}\! \left(x \right)+F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{21}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{23}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{27}\! \left(x \right)+F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{21}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{23}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{21}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{23}\! \left(x \right)+F_{64}\! \left(x \right)
F_{70}\! \left(x \right) = F_{21}\! \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_{23}\! \left(x \right)
F_{73}\! \left(x \right) = F_{64}\! \left(x \right)
F_{74}\! \left(x \right) = F_{52}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{25}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{23}\! \left(x \right) F_{25}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{21}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{69}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{21}\! \left(x \right) F_{25}\! \left(x \right) F_{81}\! \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_21(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_21(x)*F_7(x))
Eq(F_7(x), F_75(x) + F_8(x))
Eq(F_8(x), F_54(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_45(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_21(x))
Eq(F_13(x), F_14(x) + F_31(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_21(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_21(x))
Eq(F_21(x), x)
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_21(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_24(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_21(x)*F_29(x))
Eq(F_29(x), F_19(x) + F_26(x))
Eq(F_30(x), F_21(x)*F_22(x))
Eq(F_31(x), F_23(x) + F_32(x))
Eq(F_32(x), F_27(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_21(x)*F_34(x))
Eq(F_34(x), F_15(x) + F_32(x))
Eq(F_35(x), F_21(x)*F_36(x))
Eq(F_36(x), F_22(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_21(x)*F_40(x))
Eq(F_40(x), F_23(x) + F_38(x))
Eq(F_41(x), 2*F_27(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_21(x)*F_43(x))
Eq(F_43(x), F_26(x) + F_41(x))
Eq(F_44(x), F_21(x)*F_37(x))
Eq(F_45(x), F_11(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_21(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_15(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_21(x)*F_53(x))
Eq(F_53(x), F_15(x) + F_51(x))
Eq(F_54(x), F_55(x) + F_74(x))
Eq(F_55(x), F_23(x) + F_56(x))
Eq(F_56(x), F_27(x) + F_57(x) + F_70(x))
Eq(F_57(x), F_21(x)*F_58(x))
Eq(F_58(x), F_15(x) + F_59(x))
Eq(F_59(x), F_27(x) + F_57(x) + F_60(x))
Eq(F_60(x), F_21(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_23(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_27(x) + F_65(x) + F_68(x))
Eq(F_65(x), F_21(x)*F_66(x))
Eq(F_66(x), F_23(x) + F_67(x))
Eq(F_67(x), F_65(x))
Eq(F_68(x), F_21(x)*F_69(x))
Eq(F_69(x), F_23(x) + F_64(x))
Eq(F_70(x), F_21(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_23(x))
Eq(F_73(x), F_64(x))
Eq(F_74(x), F_52(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_25(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_23(x)*F_25(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_21(x)*F_81(x))
Eq(F_81(x), F_69(x) + F_82(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_21(x)*F_25(x)*F_81(x))
Pack JSON:
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Specification JSON:
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