A264229 G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-6*x) ), with A(0) = 0.
1, 2, 8, 38, 192, 1008, 5428, 29752, 165232, 926986, 5242696, 29846440, 170846760, 982496400, 5672562432, 32864292248, 190977464576, 1112761458944, 6499186961080, 38040656888144, 223089977217248, 1310627164161296, 7712227735497024, 45449101195872960, 268204421736352320, 1584740639910023552, 9374834857254623744, 55519826063209918038
Offset: 1
Keywords
Examples
G.f.: A(x) = x + 2*x^2 + 8*x^3 + 38*x^4 + 192*x^5 + 1008*x^6 + 5428*x^7 + 29752*x^8 + 165232*x^9 + 926986*x^10 + 5242696*x^11 + 29846440*x^12 + ... where A(x)^3 = A( x^3/(1-6*x) ). RELATED SERIES. A(x)^3 = x^3 + 6*x^4 + 36*x^5 + 218*x^6 + 1320*x^7 + 7992*x^8 + 48392*x^9 + 293040*x^10 + 1774656*x^11 + 10748198*x^12 + 65101584*x^13 + ... A( x/(1 + 2*x + 4*x^2) ) = x + 6*x^4 + 52*x^7 + 554*x^10 + 6888*x^13 + 95768*x^16 + 1435832*x^19 + 22605648*x^22 + 367354432*x^25 + 6097422934*x^28 + 102720725488*x^31 + 1749623396240*x^34 + 30056679361984*x^37 + ... Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where B(x) = 1 + 2*x + 4*x^2 + 6*x^3 - 24*x^5 - 56*x^6 + 368*x^8 + 986*x^9 - 7496*x^11 - 21144*x^12 + 173824*x^14 + 505040*x^15 - 4353184*x^17 + ... Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + 2*x + C2(x): C0(x) = 1 + 6*x^3 - 56*x^6 + 986*x^9 - 21144*x^12 + 505040*x^15 - 12892588*x^18 + 344317272*x^21 - 9501257152*x^24 + ... C2(x) = 4*x^2 - 24*x^5 + 368*x^8 - 7496*x^11 + 173824*x^14 - 4353184*x^17 + 114716608*x^20 - 3134509760*x^23 + ... then C0(x) = 4*x^2/C2(x).
Programs
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PARI
{a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-6*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)} for(n=1, 40, print1(a(n), ", "))
Formula
From Paul D. Hanna, Mar 17 2024: (Start)
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x)^3 = A( x^3/(1 - 6*x) ).
(2) A( x/(1 + 6*x) )^3 = A( x^3/(1 + 6*x)^2 ).
(3) A( x/(1 + 2*x + 4*x^2) )^3 = A( x^3/(1 - 8*x^3)^2 ). (End)
Comments