0132_0213_0231_2013_3120
Counting sequence:
1, 1, 2, 6, 19, 56, 150, 367, 832, 1776, 3624, 7164, 13877, 26588, 50758, 97057, 186498, 360692, 702344, 1376230, 2711135, 5363712, 10646182, 21181715, 42215172, 84234568, 168213976, 336099312, 671779881, 1343031444, 2685402326, 5369985605, 10738963462, 21476695852, 42951897816, 85901994074, 171801828179, 343601080792, 687199106198, 1374394605303, 2748784971592, 5497564983008, 10995124185672, 21990241661284, 43980475561885, 87960942179340, 175921874084262, 351843736403817, 703687459377290, 1407374903467172, 2814749789581288, 5629499559517006, 11258999096849607, 22517998168708976, 45035996309332934, 90071992587173851, 180143985139111820, 360287970238880952, 720575940433921912, 1441151880819086936, 2882303761584049809, 5764607523108125764, 11529215046149911222, 23058430092226563213, 46116860184372358158, 92233720368655809372, 184467440737213902008, 368934881474320562882, 737869762948523600059, 1475739525896918581960, 2951479051793696595510, 5902958103587239762367, 11805916207174312271312, 23611832414348442442960, 47223664828696686859112, 94447329657393158621388, 188894659314786083868421, 377789318629571914810172, 755578637259143555796486, 1511157274518286815454129, 3022314549036573310960786, 6044629098073146276592980, 12089258196146292180821832, 24178516392292583960504502, 48357032784585167489266991, 96714065569170334514269664, 193428131138340668529738214, 386856262276681336524025507, 773712524553362672473735188, 1547425049106725344331968808, 3094850098213450688004819992, 6189700196426901375304362688, 12379400392853802749854627577, 24758800785707605498903554804, 49517601571415210996946899350, 99035203142830421992976041685, 198070406285660843984973608982, 396140812571321687968904717452, 792281625142643375936699456920, 1584563250285286751872217859882, 3169126500570573503743179839523
Generating function in Maple syntax:
(x^8-4*x^7+17*x^6-45*x^5+63*x^4-54*x^3+28*x^2-8*x+1)/(2*x-1)/(x-1)^7
Generating function in latex syntax:
\frac{x^{8}-4 x^{7}+17 x^{6}-45 x^{5}+63 x^{4}-54 x^{3}+28 x^{2}-8 x +1}{\left(2 x -1\right) \left(x -1\right)^{7}}
Generating function in sympy syntax:
(x**8 - 4*x**7 + 17*x**6 - 45*x**5 + 63*x**4 - 54*x**3 + 28*x**2 - 8*x + 1)/((x - 1)**7*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^7*F(x)-x^8+4*x^7-17*x^6+45*x^5-63*x^4+54*x^3-28*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{7} F \! \left(x \right)-x^{8}+4 x^{7}-17 x^{6}+45 x^{5}-63 x^{4}+54 x^{3}-28 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-2+1/240*n^5-13/144*n^4+5/16*n^3-127/90*n^2-49/60*n+1/720*n^6+5*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -2+\frac{n^{5}}{240}-\frac{13 n^{4}}{144}+\frac{5 n^{3}}{16}-\frac{127 n^{2}}{90}-\frac{49 n}{60}+\frac{n^{6}}{720}+5 \,2^{-1+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) = 56
a(6) = 150
a(7) = 367
a(8) = 832
a(n+1) = 2*a(n)-1/720*n*(-1+n)*(n^4-2*n^3-97*n^2+338*n-1008), n >= 9
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) = 56
a \! \left(6\right) = 150
a \! \left(7\right) = 367
a \! \left(8\right) = 832
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{n \left(-1+n \right) \left(n^{4}-2 n^{3}-97 n^{2}+338 n -1008\right)}{720}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18191/
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[24,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[23,x]
F[22,x] = F[7,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]+F[26,x]+F[56,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[33,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[13,x]+F[32,x]
F[32,x] = F[29,x]
F[33,x] = F[24,x]+F[34,x]
F[34,x] = 2*F[25,x]+F[35,x]+F[39,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[29,x]
F[38,x] = F[34,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[19,x]
F[42,x] = F[43,x]
F[43,x] = 2*F[25,x]+F[44,x]+F[52,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[37,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[29,x]
F[51,x] = F[47,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[19,x]
F[55,x] = F[43,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[73,x]
F[58,x] = F[15,x]+F[59,x]
F[59,x] = F[25,x]+F[60,x]+F[72,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = F[11,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[13,x]*F[4,x]
F[65,x] = F[59,x]+F[66,x]
F[66,x] = 2*F[25,x]+F[67,x]+F[71,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[63,x]
F[70,x] = F[66,x]
F[71,x] = F[4,x]*F[41,x]
F[72,x] = F[4,x]*F[58,x]
F[73,x] = F[74,x]
F[74,x] = F[25,x]+F[75,x]+F[78,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[28,x]+F[77,x]
F[77,x] = F[19,x]+F[47,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[86,x]
F[80,x] = F[15,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[11,x]
F[85,x] = F[81,x]
F[86,x] = F[74,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_{24}\! \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_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{56}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{29}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{19}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{44}\! \left(x \right)+F_{52}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{29}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{19}\! \left(x \right)
F_{55}\! \left(x \right) = F_{43}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{73}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{25}\! \left(x \right)+F_{60}\! \left(x \right)+F_{72}\! \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_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{11}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{59}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{63}\! \left(x \right)
F_{70}\! \left(x \right) = F_{66}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{25}\! \left(x \right)+F_{75}\! \left(x \right)+F_{78}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{28}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{19}\! \left(x \right)+F_{47}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{86}\! \left(x \right)
F_{80}\! \left(x \right) = F_{15}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{11}\! \left(x \right)
F_{85}\! \left(x \right) = F_{81}\! \left(x \right)
F_{86}\! \left(x \right) = F_{74}\! \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_24(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_23(x))
Eq(F_22(x), F_7(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_56(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_33(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_13(x) + F_32(x))
Eq(F_32(x), F_29(x))
Eq(F_33(x), F_24(x) + F_34(x))
Eq(F_34(x), 2*F_25(x) + F_35(x) + F_39(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_29(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_19(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), 2*F_25(x) + F_44(x) + F_52(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_37(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_29(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_19(x))
Eq(F_55(x), F_43(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_73(x))
Eq(F_58(x), F_15(x) + F_59(x))
Eq(F_59(x), F_25(x) + F_60(x) + F_72(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), F_11(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_13(x)*F_4(x))
Eq(F_65(x), F_59(x) + F_66(x))
Eq(F_66(x), 2*F_25(x) + F_67(x) + F_71(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_63(x))
Eq(F_70(x), F_66(x))
Eq(F_71(x), F_4(x)*F_41(x))
Eq(F_72(x), F_4(x)*F_58(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_25(x) + F_75(x) + F_78(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_28(x) + F_77(x))
Eq(F_77(x), F_19(x) + F_47(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_86(x))
Eq(F_80(x), F_15(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_11(x))
Eq(F_85(x), F_81(x))
Eq(F_86(x), F_74(x))
Pack JSON:
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Specification JSON:
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