0231_1032_2130_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 181, 509, 1399, 3790, 10170, 27112, 71943, 190263, 501929, 1321680, 3475412, 9129146, 23961193, 62852753, 164793691, 431921554, 1131757422, 2964923596, 7766159115, 20339845227, 53265959501, 139483199124, 365233969544, 956319372830, 2503925475565, 6555859707077, 17164458952063, 44939127761878, 117656145559074, 308035751366320, 806463993441807, 2111381998762911, 5527733542454513, 14471921707815768, 37888237739423036, 99193203827313794, 259692198376239217, 679885040568845561, 1779966221865180835, 4660020222096463642, 12200107638563743446, 31940329081873833364, 83620932383615889939, 218922573622090102995, 573146999588886952085, 1500518847357035819292, 3928410386907150637808, 10284714002214276358118, 26925734997435398964469, 70492497745491361591085, 184551771749837567920327, 483162844525619106392926, 1264936815870215279704458, 3311647711171417789612408, 8670006533816820202916631, 22698372322624607046705159, 59425111298748129392334137, 155576963303002038040567776, 407305782069022498549910180, 1066340389821594485250244682, 2791715401230819012482987641, 7308805841540978662763045729, 19134702178732349196934804459, 50095300805336533370298677410, 131151200458638179798475847230, 343358301013299863794158103132, 898923703466705127122056939803, 2353412811157702948648129671483, 6161314733548178580974565984989, 16130531396570382518580036104100, 42230279470330068423374477968472, 110560307042754021648761269083566, 289450641714600394317345071846653, 757791618214383956892145431585173, 1983924213155225067536834193166351, 5193981021704638428073843088428774, 13598018852865384581395666953149682, 35600075538704904045535101533079616, 93202207766876105014053525170207775, 244006547769176965914313249025780847, 638817435555161902564261772003608961, 1672445758925322961449223167177994344, 4378519841278835421124909929916270604, 11463113765027240180608511023342610450, 30010821454034998878066631941655146625, 78569350597541983968323402404710001097, 205697230339519408056367610478649199923, 538522340422873150259707499443586285098, 1409869790932813862840611028676807028134, 3691087032382996078497837868236229544740, 9663391306231029653124327139330671096867, 25299086886339803441817992676353362727331, 66233869352847801794215349142924575047973, 173402521172322444184599451258810678342188, 453973694164357215247125797646288091829696
Generating function in Maple syntax:
-(x^6-3*x^5+4*x^4-11*x^3+13*x^2-6*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{6}-3 x^{5}+4 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**6 + 3*x**5 - 4*x**4 + 11*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^2*F(x)+x^6-3*x^5+4*x^4-11*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{6}-3 x^{5}+4 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(2*5^(1/2)+10)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-2*5^(1/2)+10)*(3/2+1/2*5^(1/2))^(-n)-n-3/4*2^n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(2 \sqrt{5}+10\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-2 \sqrt{5}+10\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}-n -\frac{3 \,2^{n}}{4}+2 & \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) = 62
a(6) = 181
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)-n+2, n >= 7
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) = 62
a \! \left(6\right) = 181
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-n +2, \quad n \geq 7
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21450/
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[9,x]
F[3,x] = F[1,x]+F[4,x]+F[43,x]
F[4,x] = F[5,x]*F[9,x]
F[5,x] = F[1,x]+F[32,x]+F[4,x]+F[6,x]
F[6,x] = F[7,x]*F[9,x]
F[7,x] = F[1,x]+F[10,x]+F[12,x]+F[31,x]+F[8,x]
F[8,x] = F[7,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]*F[9,x]
F[11,x] = F[1,x]+F[10,x]+F[12,x]+F[30,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[17,x]*F[9,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[9,x]
F[17,x] = F[1,x]+F[18,x]+F[19,x]
F[18,x] = F[11,x]*F[9,x]
F[19,x] = F[20,x]*F[9,x]
F[20,x] = F[14,x]+F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[9,x]
F[24,x] = F[1,x]+F[22,x]
F[25,x] = F[26,x]+F[27,x]+F[29,x]
F[26,x] = 0
F[27,x] = F[28,x]*F[9,x]
F[28,x] = F[15,x]+F[25,x]
F[29,x] = F[21,x]*F[9,x]
F[30,x] = F[11,x]*F[9,x]
F[31,x] = F[7,x]*F[9,x]
F[32,x] = F[33,x]*F[9,x]
F[33,x] = F[20,x]+F[34,x]
F[34,x] = F[21,x]+F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[9,x]
F[38,x] = F[22,x]+F[36,x]
F[39,x] = 2*F[26,x]+F[40,x]+F[42,x]
F[40,x] = F[41,x]*F[9,x]
F[41,x] = F[25,x]+F[39,x]
F[42,x] = F[35,x]*F[9,x]
F[43,x] = F[44,x]*F[9,x]
F[44,x] = F[1,x]+F[45,x]+F[46,x]+F[56,x]
F[45,x] = F[44,x]*F[9,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]*F[9,x]
F[48,x] = F[49,x]+F[55,x]
F[49,x] = F[1,x]+F[50,x]+F[51,x]
F[50,x] = F[49,x]*F[9,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]*F[9,x]
F[53,x] = F[49,x]+F[54,x]
F[54,x] = F[14,x]*F[49,x]*F[9,x]
F[55,x] = F[24,x]*F[9,x]
F[56,x] = F[57,x]*F[9,x]
F[57,x] = F[1,x]+F[46,x]+F[56,x]+F[58,x]+F[62,x]
F[58,x] = F[59,x]
F[59,x] = F[60,x]*F[9,x]
F[60,x] = F[44,x]+F[61,x]
F[61,x] = F[14,x]*F[24,x]*F[9,x]
F[62,x] = F[63,x]
F[63,x] = F[14,x]*F[64,x]*F[9,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[14,x]*F[66,x]
F[66,x] = F[1,x]+F[67,x]
F[67,x] = F[49,x]*F[9,x]
F[68,x] = F[69,x]
F[69,x] = F[70,x]*F[9,x]
F[70,x] = F[54,x]+F[68,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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{43}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{4}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{31}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{9}\! \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_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right) F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{31}\! \left(x \right) = F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{9}\! \left(x \right)
F_{33}\! \left(x \right) = F_{20}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{21}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{9}\! \left(x \right)
F_{38}\! \left(x \right) = F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{9}\! \left(x \right)
F_{41}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right) F_{9}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right) F_{9}\! \left(x \right)
F_{44}\! \left(x \right) = F_{1}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \left(x \right)
F_{45}\! \left(x \right) = F_{44}\! \left(x \right) F_{9}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{9}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{49}\! \left(x \right) = F_{1}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{49}\! \left(x \right) F_{9}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right) F_{9}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right) F_{49}\! \left(x \right) F_{9}\! \left(x \right)
F_{55}\! \left(x \right) = F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right) F_{9}\! \left(x \right)
F_{57}\! \left(x \right) = F_{1}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right) F_{9}\! \left(x \right)
F_{60}\! \left(x \right) = F_{44}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{14}\! \left(x \right) F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{14}\! \left(x \right) F_{64}\! \left(x \right) F_{9}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{14}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{1}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{49}\! \left(x \right) F_{9}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right) F_{9}\! \left(x \right)
F_{70}\! \left(x \right) = F_{54}\! \left(x \right)+F_{68}\! \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)*F_9(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_43(x))
Eq(F_4(x), F_5(x)*F_9(x))
Eq(F_5(x), F_1(x) + F_32(x) + F_4(x) + F_6(x))
Eq(F_6(x), F_7(x)*F_9(x))
Eq(F_7(x), F_1(x) + F_10(x) + F_12(x) + F_31(x) + F_8(x))
Eq(F_8(x), F_7(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x)*F_9(x))
Eq(F_11(x), F_1(x) + F_10(x) + F_12(x) + F_30(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_17(x)*F_9(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_9(x))
Eq(F_17(x), F_1(x) + F_18(x) + F_19(x))
Eq(F_18(x), F_11(x)*F_9(x))
Eq(F_19(x), F_20(x)*F_9(x))
Eq(F_20(x), F_14(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_9(x))
Eq(F_24(x), F_1(x) + F_22(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_29(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_28(x)*F_9(x))
Eq(F_28(x), F_15(x) + F_25(x))
Eq(F_29(x), F_21(x)*F_9(x))
Eq(F_30(x), F_11(x)*F_9(x))
Eq(F_31(x), F_7(x)*F_9(x))
Eq(F_32(x), F_33(x)*F_9(x))
Eq(F_33(x), F_20(x) + F_34(x))
Eq(F_34(x), F_21(x) + F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_9(x))
Eq(F_38(x), F_22(x) + F_36(x))
Eq(F_39(x), 2*F_26(x) + F_40(x) + F_42(x))
Eq(F_40(x), F_41(x)*F_9(x))
Eq(F_41(x), F_25(x) + F_39(x))
Eq(F_42(x), F_35(x)*F_9(x))
Eq(F_43(x), F_44(x)*F_9(x))
Eq(F_44(x), F_1(x) + F_45(x) + F_46(x) + F_56(x))
Eq(F_45(x), F_44(x)*F_9(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x)*F_9(x))
Eq(F_48(x), F_49(x) + F_55(x))
Eq(F_49(x), F_1(x) + F_50(x) + F_51(x))
Eq(F_50(x), F_49(x)*F_9(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x)*F_9(x))
Eq(F_53(x), F_49(x) + F_54(x))
Eq(F_54(x), F_14(x)*F_49(x)*F_9(x))
Eq(F_55(x), F_24(x)*F_9(x))
Eq(F_56(x), F_57(x)*F_9(x))
Eq(F_57(x), F_1(x) + F_46(x) + F_56(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_60(x)*F_9(x))
Eq(F_60(x), F_44(x) + F_61(x))
Eq(F_61(x), F_14(x)*F_24(x)*F_9(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_14(x)*F_64(x)*F_9(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_14(x)*F_66(x))
Eq(F_66(x), F_1(x) + F_67(x))
Eq(F_67(x), F_49(x)*F_9(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_70(x)*F_9(x))
Eq(F_70(x), F_54(x) + F_68(x))
Pack JSON:
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Specification JSON:
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