0132_1032_1203_2310
Counting sequence:
1, 1, 2, 6, 20, 61, 166, 416, 988, 2267, 5086, 11238, 24564, 53257, 114726, 245836, 524412, 1114295, 2359550, 4981074, 10486196, 22020645, 46138022, 96469816, 201327580, 419431571, 872416606, 1811940926, 3758098228, 7784630337, 16106129766, 33285999268, 68719479804, 141733924207, 292057779966, 601295425706, 1236950585972, 2542620644445, 5222680237670, 10720238377104, 21990232562396, 45079976746315, 92358976741342, 189115999986326, 387028092986740, 791648372009081, 1618481116097446, 3307330976362236, 6755399441068668, 13792273858836007, 28147497671080446, 57420895248989698, 117093590311649844, 238690780250654357, 486388759756032806, 990791918021529576, 2017612633062003932, 4107282860161915395, 8358680908399664990, 17005592192951018734, 34587645138205436596, 70328211781017694385, 142962266571249055462, 290536219160925470036, 590295810358705685500, 1199038364791120890591, 2434970217729660850686, 4943727411754159872346, 10035028776097996120308, 20365205457375345027277, 41320706725109395665126, 83822005070936202590528, 170005193383307227742684, 344732753249484100651707, 698910239464707491681246, 1416709944860893564165446, 2871198821584744289986292, 5817955506895402903335145, 11787026741242634453449510, 23876284937388926200513964, 48357032784585166988316796, 97922991388784963151272855, 198263834416799184651888382, 401363372112056886002528946, 812398150781030805402631860, 1644139114675895677600484101, 3326963855579459488791484326, 6731298963614255244764079256, 13617340432139183023890461148, 27544165874099711116505612147, 55707301767842112370460691806, 112652543574969605015820409758, 227780967228509970581438966324, 460513694614161462262474324257, 930930909542605966724141533286, 1881668859713778017846668941316, 3802951800684688204490109741052, 7685131763883640746573763311695, 15528719852795810168334614399230, 31374352355648677687043404470794, 63382530011411470074835160410996
Generating function in Maple syntax:
(4*x^7-10*x^5+21*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{4 x^{7}-10 x^{5}+21 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(4*x**7 - 10*x**5 + 21*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4*F(x)+4*x^7-10*x^5+21*x^4-28*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+4 x^{7}-10 x^{5}+21 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-4+1/6*n^3-5/2*n^2+16/3*n+2^(-1+n)*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -4+\frac{n^{3}}{6}-\frac{5 n^{2}}{2}+\frac{16 n}{3}+2^{-1+n} 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) = 20
a(5) = 61
a(6) = 166
a(7) = 416
a(n+2) = 1/6*n^3-7/2*n^2-4*a(n)+4*a(n+1)+46/3*n-14, 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) = 61
a \! \left(6\right) = 166
a \! \left(7\right) = 416
a \! \left(n +2\right) = \frac{n^{3}}{6}-\frac{7 n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{46 n}{3}-14, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15506/
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[32,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[26,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[10,x]+F[21,x]
F[25,x] = F[12,x]*F[17,x]
F[26,x] = F[22,x]+F[27,x]+F[31,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[21,x]
F[31,x] = F[12,x]*F[13,x]
F[32,x] = F[33,x]+F[49,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[1,x]+F[14,x]
F[37,x] = F[18,x]+F[38,x]
F[38,x] = F[22,x]+F[23,x]+F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[17,x]+F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[18,x]+F[42,x]
F[45,x] = 2*F[22,x]+F[46,x]+F[48,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[21,x]+F[45,x]
F[48,x] = F[12,x]*F[41,x]
F[49,x] = F[22,x]+F[50,x]+F[60,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[56,x]
F[52,x] = F[10,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[14,x]+F[53,x]
F[56,x] = F[21,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[38,x]+F[57,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = F[33,x]+F[63,x]
F[63,x] = F[22,x]+F[50,x]+F[64,x]
F[64,x] = F[12,x]*F[62,x]
F[65,x] = F[66,x]+F[79,x]
F[66,x] = F[22,x]+F[39,x]+F[67,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]+F[74,x]
F[69,x] = F[10,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[18,x]+F[73,x]
F[73,x] = F[71,x]
F[74,x] = F[21,x]+F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[42,x]+F[78,x]
F[78,x] = F[76,x]
F[79,x] = F[22,x]+F[80,x]+F[81,x]+F[84,x]
F[80,x] = 0
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[30,x]+F[83,x]
F[83,x] = F[45,x]
F[84,x] = F[12,x]*F[65,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_{32}\! \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_{26}\! \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_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \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_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{49}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \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_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{49}\! \left(x \right) = F_{22}\! \left(x \right)+F_{50}\! \left(x \right)+F_{60}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{33}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{22}\! \left(x \right)+F_{50}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{79}\! \left(x \right)
F_{66}\! \left(x \right) = F_{22}\! \left(x \right)+F_{39}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{74}\! \left(x \right)
F_{69}\! \left(x \right) = F_{10}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{71}\! \left(x \right)
F_{74}\! \left(x \right) = F_{21}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{42}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{22}\! \left(x \right)+F_{80}\! \left(x \right)+F_{81}\! \left(x \right)+F_{84}\! \left(x \right)
F_{80}\! \left(x \right) = 0
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{30}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{45}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \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_32(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_26(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_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_10(x) + F_21(x))
Eq(F_25(x), F_12(x)*F_17(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_31(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_21(x))
Eq(F_31(x), F_12(x)*F_13(x))
Eq(F_32(x), F_33(x) + F_49(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_1(x) + F_14(x))
Eq(F_37(x), F_18(x) + F_38(x))
Eq(F_38(x), F_22(x) + F_23(x) + F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_17(x) + F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_18(x) + F_42(x))
Eq(F_45(x), 2*F_22(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_21(x) + F_45(x))
Eq(F_48(x), F_12(x)*F_41(x))
Eq(F_49(x), F_22(x) + F_50(x) + F_60(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_56(x))
Eq(F_52(x), F_10(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_14(x) + F_53(x))
Eq(F_56(x), F_21(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_38(x) + F_57(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), F_33(x) + F_63(x))
Eq(F_63(x), F_22(x) + F_50(x) + F_64(x))
Eq(F_64(x), F_12(x)*F_62(x))
Eq(F_65(x), F_66(x) + F_79(x))
Eq(F_66(x), F_22(x) + F_39(x) + F_67(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_74(x))
Eq(F_69(x), F_10(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_18(x) + F_73(x))
Eq(F_73(x), F_71(x))
Eq(F_74(x), F_21(x) + F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_42(x) + F_78(x))
Eq(F_78(x), F_76(x))
Eq(F_79(x), F_22(x) + F_80(x) + F_81(x) + F_84(x))
Eq(F_80(x), 0)
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_30(x) + F_83(x))
Eq(F_83(x), F_45(x))
Eq(F_84(x), F_12(x)*F_65(x))
Pack JSON:
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Specification JSON:
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