0132_1320_2130_3120
Counting sequence:
1, 1, 2, 6, 20, 64, 188, 509, 1291, 3117, 7257, 16452, 36578, 80166, 173826, 373863, 799045, 1699211, 3598631, 7595014, 15982100, 33543996, 70242152, 146785921, 306166895, 637514009, 1325376613, 2751436344, 5704222326, 11811124482, 24427585982, 50465819787, 104152905033, 214748306391, 442381565971, 910532993498, 1872605659400, 3848290606456, 7902739724036, 16217796398469, 33260226617555, 68169720787141, 139637976579377, 285873023059436, 584940185799370, 1196268650826654, 2445313859962938, 4996180836384879, 10203467905512461, 20829148276318947, 42502721483017407, 86694292826567022, 176766285373952188, 360287970189273716, 734086739260997216, 1495195076286581833, 3044433348102001655, 6196953087261316401, 12610078956636868541, 25652503477501789024, 52169698083459231710, 106068778423829288186, 215596321361479709302, 438110171750601132051, 890055401556485100945, 1807780919223535245999, 3670902070668199909163, 7452484605778657938114, 15126330140441831355696, 30695382138652692862512, 62276207992843445169532, 126323303416763008318157, 256188381695678251630043, 519460313115660972226141, 1053087725679930881303561, 2134509650257079635166868, 4325687698308595014245810, 8764712192206061515041046, 17756097975589866001836146, 35965543133535217945762679, 72837780631781407774212565, 147488949992984759312227131, 298604677444813406150403927, 604462909807314587350967766, 1223432929450004724800427620, 2475880078570760549795920044, 5009788596483023299979955288, 10135634071649051000734028049, 20503381900664110803014076031, 41470991316060239209117871177, 83870437661584513624412750357, 169597785382097097661176973192, 342909390882050336147054230598, 693246421999812953943506247666, 1401348124471050471185805161006, 2832406809884950068969192616603, 5724234741655598391133546651865, 11567311727082593288657412832391, 23372307941707979590095461270851, 47219984858501545205752190155434, 95390707667174262462626911788120
Generating function in Maple syntax:
-(x^7+5*x^6-28*x^5+49*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}+5 x^{6}-28 x^{5}+49 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 - 5*x**6 + 28*x**5 - 49*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+5*x^6-28*x^5+49*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}+5 x^{6}-28 x^{5}+49 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-7/4*n+1/24*n^2+1-1/4*n^3-1/24*n^4+3/4*2^n*n+1/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{7 n}{4}+\frac{n^{2}}{24}+1-\frac{n^{3}}{4}-\frac{n^{4}}{24}+\frac{3 \,2^{n} n}{4}+\frac{2^{n}}{4} & \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) = 64
a(6) = 188
a(7) = 509
a(n+2) = -1/24*n^4+1/12*n^3+37/24*n^2-4*a(n)+4*a(n+1)-31/12*n+3, 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) = 64
a \! \left(6\right) = 188
a \! \left(7\right) = 509
a \! \left(n +2\right) = -\frac{n^{4}}{24}+\frac{n^{3}}{12}+\frac{37 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{31 n}{12}+3, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15677/
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[27,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[18,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[14,x]
F[18,x] = F[19,x]+F[20,x]+F[26,x]
F[19,x] = 0
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[10,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[22,x]
F[25,x] = F[18,x]
F[26,x] = F[12,x]*F[13,x]
F[27,x] = F[14,x]+F[28,x]
F[28,x] = F[19,x]+F[29,x]+F[54,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[53,x]
F[31,x] = F[32,x]+F[6,x]
F[32,x] = F[19,x]+F[24,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[10,x]+F[36,x]
F[36,x] = F[19,x]+F[24,x]+F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[19,x]+F[24,x]+F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[10,x]
F[43,x] = F[39,x]
F[44,x] = F[19,x]+F[45,x]+F[48,x]+F[52,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[23,x]+F[47,x]
F[47,x] = F[45,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[36,x]
F[51,x] = F[44,x]
F[52,x] = F[12,x]*F[38,x]
F[53,x] = F[28,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[63,x]
F[56,x] = F[14,x]+F[57,x]
F[57,x] = F[19,x]+F[58,x]+F[62,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[10,x]+F[36,x]
F[61,x] = F[57,x]
F[62,x] = F[12,x]*F[56,x]
F[63,x] = F[64,x]+F[70,x]
F[64,x] = F[19,x]+F[65,x]+F[67,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[22,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[17,x]+F[69,x]
F[69,x] = F[64,x]
F[70,x] = F[19,x]+F[71,x]+F[77,x]+F[81,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[36,x]+F[74,x]
F[74,x] = 2*F[19,x]+F[45,x]+F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[23,x]+F[74,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[57,x]
F[80,x] = F[70,x]
F[81,x] = F[12,x]*F[63,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_{27}\! \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_{18}\! \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_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)+F_{54}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{53}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{23}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \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_{36}\! \left(x \right)
F_{51}\! \left(x \right) = F_{44}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{53}\! \left(x \right) = F_{28}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{63}\! \left(x \right)
F_{56}\! \left(x \right) = F_{14}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{19}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{70}\! \left(x \right)
F_{64}\! \left(x \right) = F_{19}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{22}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{17}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{64}\! \left(x \right)
F_{70}\! \left(x \right) = F_{19}\! \left(x \right)+F_{71}\! \left(x \right)+F_{77}\! \left(x \right)+F_{81}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{36}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{23}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = F_{57}\! \left(x \right)
F_{80}\! \left(x \right) = F_{70}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \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_27(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_18(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_14(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_26(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_10(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_22(x))
Eq(F_25(x), F_18(x))
Eq(F_26(x), F_12(x)*F_13(x))
Eq(F_27(x), F_14(x) + F_28(x))
Eq(F_28(x), F_19(x) + F_29(x) + F_54(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_53(x))
Eq(F_31(x), F_32(x) + F_6(x))
Eq(F_32(x), F_19(x) + F_24(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_10(x) + F_36(x))
Eq(F_36(x), F_19(x) + F_24(x) + F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_19(x) + F_24(x) + F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_10(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_19(x) + F_45(x) + F_48(x) + F_52(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_23(x) + F_47(x))
Eq(F_47(x), F_45(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_36(x))
Eq(F_51(x), F_44(x))
Eq(F_52(x), F_12(x)*F_38(x))
Eq(F_53(x), F_28(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_63(x))
Eq(F_56(x), F_14(x) + F_57(x))
Eq(F_57(x), F_19(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_10(x) + F_36(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_12(x)*F_56(x))
Eq(F_63(x), F_64(x) + F_70(x))
Eq(F_64(x), F_19(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_22(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_17(x) + F_69(x))
Eq(F_69(x), F_64(x))
Eq(F_70(x), F_19(x) + F_71(x) + F_77(x) + F_81(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_36(x) + F_74(x))
Eq(F_74(x), 2*F_19(x) + F_45(x) + F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_23(x) + F_74(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), F_57(x))
Eq(F_80(x), F_70(x))
Eq(F_81(x), F_12(x)*F_63(x))
Pack JSON:
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Specification JSON:
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