0123_0132_1203_1230_2013
Counting sequence:
1, 1, 2, 6, 19, 61, 198, 651, 2171, 7345, 25194, 87516, 307462, 1091026, 3905580, 14088665, 51165715, 186923705, 686483650, 2532947000, 9385112530, 34905108470, 130262393140, 487635569270, 1830633108574, 6890248857626, 25996000238628, 98296251919656, 372439551601516, 1413839817188036, 5376675968066648, 20480759256616533, 78135960848962051, 298529351193892105, 1142132549248223666, 4375270420841774720, 16781109690486970906, 64436939824827929422, 247697139726180364388, 953134984070708085402, 3671237775900925777834, 14153841729433281780478, 54616067736856617277356, 210927266175256244122152, 815256742826519385712404, 3153477740932204799929788, 12206866219727338223564136, 47285059631035301870257722, 183288915035515748942031534, 710933403251288223574250250, 2759248320836950597239674484, 10715468409252439788031782096, 41637056436007028487351391092, 161878031711509548889883394396, 629688743049417509026680034824, 2450674513223917983347676590316, 9542431619655937121804922259548, 37173944551633033779312927091476, 144882939607768646190329293618056, 564921461197217122173558490878480, 2203660431573854997269747998616664, 8599652862008868616609160622043656, 33573064543882388365755926164105392, 131120113944906274043248697881101261, 512283809463811600951859594983096035, 2002211171643954231509264656550130345, 7828206184821484869660946081991993490, 30617037233888254798814039177947926480, 119786591860219965125461250699690866410, 468806037209739102618162101576682299070, 1835329756330584214443296946224834409540, 7187321826376775742258808115799446366130, 28154542009396563517537732428668238497010, 110320113069514999453935763727436039136470, 432396724695346542341270582773077302389980, 1695228736144891197532557358402958332659240, 6647973869911604173427203911674009861524740, 26077285881839410198908622172032738324483980, 102316263718038090370967648711230396234029000, 401544222272998737161317630318216685299864590, 1576252697706536388340294139756962730763615290, 6188985786278792087757912853985900421094879150, 24305906960238704255856092051376655838418723900, 95477379987652505902401854322000169999926764880, 375131184866303158290822979380584603641143369820, 1474204180482704839818152101480808452441733657620, 5794577280560265460149320909799879338168443944600, 22781035619307159146880054294148260902552623565300, 89579936916155281132672905285508909665990440682340, 352315456715883017621733468902950398812479127735340, 1385908095342357106278416110077454833438190438133880, 5452769665164363319521333101602875665201404094314160, 21457450318436256578823740343684203188280246546489320, 84453107116137034816563863731656362526742259192557240, 332451325392518769643452613542181455935781685838070480, 1308922226709048326219132757334996136810543573515377410, 5154320514328545824980614408970822613487082258686315470, 20300153382403284493561630252764491732140382012658787690, 79964292273872396946730212725535560376879034001795736980, 315036127916243210336075387392060085614066064029476155200, 1241340014058787209915313603854489093358907235371400561380
Generating function in Maple syntax:
((-x^2-x+1)*(1-4*x)^(1/2)+3*x^2+x-1)/(4*x^2-2*x)
Generating function in latex syntax:
\frac{\left(-x^{2}-x +1\right) \sqrt{1-4 x}+3 x^{2}+x -1}{4 x^{2}-2 x}
Generating function in sympy syntax:
(3*x**2 + x + sqrt(1 - 4*x)*(-x**2 - x + 1) - 1)/(4*x**2 - 2*x)
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*F(x)^2-(2*x-1)*(3*x^2+x-1)*F(x)+x^4+4*x^3-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(3 x^{2}+x -1\right) F \! \left(x \right)+x^{4}+4 x^{3}-3 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+4) = 4*(-1+2*n)/(n+5)*a(n)+2*(6+n)/(n+5)*a(1+n)-13*(n+3)/(n+5)*a(n+2)+(27+7*n)/(n+5)*a(n+3), n >= 5
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(n +4\right) = \frac{4 \left(-1+2 n \right) a \! \left(n \right)}{n +5}+\frac{2 \left(6+n \right) a \! \left(1+n \right)}{n +5}-\frac{13 \left(n +3\right) a \! \left(n +2\right)}{n +5}+\frac{\left(27+7 n \right) a \! \left(n +3\right)}{n +5}, \quad n \geq 5
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21510/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[11,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[60,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[7,x,k[0]]+F[9,x,k[0]]
F[7,x,k[0]] = F[6,x,k[0]]*F[8,x,k[0]]
F[8,x,k[0]] = k[0]*x
F[9,x,k[0]] = F[10,x,k[0]]*F[11,x]
F[10,x,k[0]] = (F[6,x,k[0]]*k[0]-F[6,x,1])/(-1+k[0])
F[11,x] = x
F[12,x,k[0]] = F[13,x,k[0]]
F[13,x,k[0]] = F[11,x]*F[14,x,k[0]]
F[14,x,k[0]] = (F[15,x,k[0]]*k[0]-F[15,x,1])/(-1+k[0])
F[15,x,k[0]] = F[16,x,k[0]]+F[58,x,k[0]]
F[16,x,k[0]] = F[17,x]+F[30,x,k[0]]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[20,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[1,x]+F[11,x]
F[22,x] = F[18,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[29,x]
F[24,x] = 0
F[25,x] = F[11,x]*F[26,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[11,x]
F[28,x] = F[23,x]
F[29,x] = F[11,x]*F[18,x]
F[30,x,k[0]] = F[31,x,k[0]]+F[34,x,k[0]]
F[31,x,k[0]] = F[32,x,k[0]]
F[32,x,k[0]] = F[33,x,k[0]]*F[8,x,k[0]]
F[33,x,k[0]] = F[1,x]+F[31,x,k[0]]
F[34,x,k[0]] = F[24,x]+F[35,x,k[0]]+F[37,x,k[0]]
F[35,x,k[0]] = F[36,x,k[0]]*F[8,x,k[0]]
F[36,x,k[0]] = F[18,x]+F[34,x,k[0]]
F[37,x,k[0]] = F[11,x]*F[38,x,k[0]]
F[38,x,k[0]] = F[39,x,k[0]]+F[42,x,k[0]]
F[39,x,k[0]] = F[31,x,k[0]]+F[40,x,k[0]]
F[40,x,k[0]] = F[41,x,k[0]]
F[41,x,k[0]] = F[11,x]*F[31,x,k[0]]
F[42,x,k[0]] = F[43,x,k[0]]+F[52,x,k[0]]
F[43,x,k[0]] = F[24,x]+F[37,x,k[0]]+F[44,x,k[0]]
F[44,x,k[0]] = F[45,x,k[0]]*F[8,x,k[0]]
F[45,x,k[0]] = F[46,x]+F[49,x,k[0]]
F[46,x] = F[47,x]
F[47,x] = F[11,x]*F[48,x]
F[48,x] = F[1,x]+F[46,x]
F[49,x,k[0]] = F[24,x]+F[44,x,k[0]]+F[50,x,k[0]]
F[50,x,k[0]] = F[11,x]*F[51,x,k[0]]
F[51,x,k[0]] = F[31,x,k[0]]+F[49,x,k[0]]
F[52,x,k[0]] = 2*F[24,x]+F[53,x,k[0]]+F[57,x,k[0]]
F[53,x,k[0]] = F[11,x]*F[54,x,k[0]]
F[54,x,k[0]] = F[55,x,k[0]]+F[56,x,k[0]]
F[55,x,k[0]] = F[40,x,k[0]]
F[56,x,k[0]] = F[52,x,k[0]]
F[57,x,k[0]] = F[11,x]*F[43,x,k[0]]
F[58,x,k[0]] = F[59,x,k[0]]
F[59,x,k[0]] = F[11,x]*F[17,x]*F[33,x,k[0]]*F[48,x]
F[60,x] = F[61,x]
F[61,x] = F[11,x]*F[62,x]
F[62,x] = F[15,x,1]
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_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{60}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{7}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{6}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = y x
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{10}\! \left(x , y\right) = \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}
F_{11}\! \left(x \right) = x
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = \frac{y F_{15}\! \left(x , y\right)-F_{15}\! \left(x , 1\right)}{-1+y}
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right)+F_{58}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{30}\! \left(x , y\right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{11}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{18}\! \left(x \right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x , y\right)+F_{34}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{33}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{37}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{36}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{18}\! \left(x \right)+F_{34}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{38}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{39}\! \left(x , y\right)+F_{42}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{31}\! \left(x , y\right)+F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{31}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{43}\! \left(x , y\right)+F_{52}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{37}\! \left(x , y\right)+F_{44}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{45}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{46}\! \left(x \right)+F_{49}\! \left(x , y\right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{1}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x , y\right) = F_{24}\! \left(x \right)+F_{44}\! \left(x , y\right)+F_{50}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{51}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{31}\! \left(x , y\right)+F_{49}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = 2 F_{24}\! \left(x \right)+F_{53}\! \left(x , y\right)+F_{57}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{55}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{40}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{52}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{43}\! \left(x , y\right)
F_{58}\! \left(x , y\right) = F_{59}\! \left(x , y\right)
F_{59}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{33}\! \left(x , y\right) F_{48}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x , 1\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_11(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_60(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_12(x, k_0) + F_7(x, k_0) + F_9(x, k_0))
Eq(F_7(x, k_0), F_6(x, k_0)*F_8(x, k_0))
Eq(F_8(x, k_0), k_0*x)
Eq(F_9(x, k_0), F_10(x, k_0)*F_11(x))
Eq(F_10(x, k_0), (-k_0*F_6(x, k_0) + F_6(x, 1))/(1 - k_0))
Eq(F_11(x), x)
Eq(F_12(x, k_0), F_13(x, k_0))
Eq(F_13(x, k_0), F_11(x)*F_14(x, k_0))
Eq(F_14(x, k_0), (-k_0*F_15(x, k_0) + F_15(x, 1))/(1 - k_0))
Eq(F_15(x, k_0), F_16(x, k_0) + F_58(x, k_0))
Eq(F_16(x, k_0), F_17(x) + F_30(x, k_0))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_1(x) + F_11(x))
Eq(F_22(x), F_18(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_29(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_11(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_11(x))
Eq(F_28(x), F_23(x))
Eq(F_29(x), F_11(x)*F_18(x))
Eq(F_30(x, k_0), F_31(x, k_0) + F_34(x, k_0))
Eq(F_31(x, k_0), F_32(x, k_0))
Eq(F_32(x, k_0), F_33(x, k_0)*F_8(x, k_0))
Eq(F_33(x, k_0), F_1(x) + F_31(x, k_0))
Eq(F_34(x, k_0), F_24(x) + F_35(x, k_0) + F_37(x, k_0))
Eq(F_35(x, k_0), F_36(x, k_0)*F_8(x, k_0))
Eq(F_36(x, k_0), F_18(x) + F_34(x, k_0))
Eq(F_37(x, k_0), F_11(x)*F_38(x, k_0))
Eq(F_38(x, k_0), F_39(x, k_0) + F_42(x, k_0))
Eq(F_39(x, k_0), F_31(x, k_0) + F_40(x, k_0))
Eq(F_40(x, k_0), F_41(x, k_0))
Eq(F_41(x, k_0), F_11(x)*F_31(x, k_0))
Eq(F_42(x, k_0), F_43(x, k_0) + F_52(x, k_0))
Eq(F_43(x, k_0), F_24(x) + F_37(x, k_0) + F_44(x, k_0))
Eq(F_44(x, k_0), F_45(x, k_0)*F_8(x, k_0))
Eq(F_45(x, k_0), F_46(x) + F_49(x, k_0))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_11(x)*F_48(x))
Eq(F_48(x), F_1(x) + F_46(x))
Eq(F_49(x, k_0), F_24(x) + F_44(x, k_0) + F_50(x, k_0))
Eq(F_50(x, k_0), F_11(x)*F_51(x, k_0))
Eq(F_51(x, k_0), F_31(x, k_0) + F_49(x, k_0))
Eq(F_52(x, k_0), 2*F_24(x) + F_53(x, k_0) + F_57(x, k_0))
Eq(F_53(x, k_0), F_11(x)*F_54(x, k_0))
Eq(F_54(x, k_0), F_55(x, k_0) + F_56(x, k_0))
Eq(F_55(x, k_0), F_40(x, k_0))
Eq(F_56(x, k_0), F_52(x, k_0))
Eq(F_57(x, k_0), F_11(x)*F_43(x, k_0))
Eq(F_58(x, k_0), F_59(x, k_0))
Eq(F_59(x, k_0), F_11(x)*F_17(x)*F_33(x, k_0)*F_48(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_11(x)*F_62(x))
Eq(F_62(x), F_15(x, 1))
Pack JSON:
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Specification JSON:
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