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%I #12 Aug 01 2019 00:20:19
%S 1,3,27,312,4140,58806,876492,13524300,214168536,3460901967,
%T 56842100298,946076020551,15922147355532,270496012834332,
%U 4632597495220104,79896692540736729,1386424262414762046,24188862129358547349,424059773742487363743,7466416997545500727257,131972899585564980561060
%N G.f. A(x) satisfies: A( 3*A(x)^3 - 27*A(x)^4 ) = 3*x^3.
%H Paul D. Hanna, <a href="/A291315/b291315.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies: A( ( A(3*x^3 - 27*x^4)/3 )^(1/3) ) = x.
%F a(n) ~ c * d^n / n^(3/2), where d = 19.04051967708439478291588279223719475817126... and c = 0.016611712761810470376477734... - _Vaclav Kotesovec_, Aug 28 2017
%e G.f.: A(x) = x + 3*x^2 + 27*x^3 + 312*x^4 + 4140*x^5 + 58806*x^6 + 876492*x^7 + 13524300*x^8 + 214168536*x^9 + 3460901967*x^10 + 56842100298*x^11 + 946076020551*x^12 + 15922147355532*x^13 + 270496012834332*x^14 + 4632597495220104*x^15 + 79896692540736729*x^16 +...
%e such that A( 3*A(x)^3 - 27*A(x)^4 ) = 3*x^3.
%e RELATED SERIES.
%e 3*A(x)^3 - 27*A(x)^4 = 3*x^3 - 27*x^6 - 243*x^9 - 3402*x^12 - 74358*x^15 - 1259712*x^18 - 26886978*x^21 - 603539829*x^24 - 13199400117*x^27 - 308337816672*x^30 - 4115921019796122114804558073934281011*x^33 +...
%e Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:
%e Ai(x) = x - 3*x^2 - 9*x^3 - 42*x^4 - 306*x^5 - 1728*x^6 - 12294*x^7 - 91989*x^8 - 670599*x^9 - 5221728*x^10 - 40781043*x^11 - 321265359*x^12 - 2579360382*x^13 - 20813948649*x^14 - 169435295856*x^15 - 1390313185839*x^16 - 11466890654004*x^17 - 95118137894619*x^18 - 792749879512335*x^19 - 6633852028922394*x^20 +...
%e where Ai(x) = ( A(3*x^3 - 27*x^4)/3 )^(1/3)
%e and Ai( 3*Ai(x)^3 ) = 3*x^3 - 27*x^4.
%o (PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = x*Ser(V); V[#V] = -polcoeff(subst(G=A, x, 3*A^3 - 27*A^4 ), #V+2)/9); V[n]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A271961, A291313, A291314.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 22 2017
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