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%I #7 Oct 13 2016 00:00:06
%S 1,6,230,11676,663174,41413812,2680851420,188202575736,11594488853190,
%T 1403241101718852,-168694608332762892,147980852726994346248,
%U -105389748185249598629220,93603955226272690108355208,-97621599698823169573607867400,118585871916859034002263802974192,-165925215611130932978517186291637434,264930310299987157013624532678164003556,-478773962141712793295304310624498264046460
%N G.f. A(x) satisfies: Series_Reversion( A(x) + 2*A(x)^3 ) = A(x) - 2*A(x)^3.
%F G.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3) satisfies:
%F (1) A( A(x) + 2*A(x)^3 ) = G(x),
%F (2) G( A(x) - 2*A(x)^3 ) = A(x),
%F (3) A( A(x) - 2*A(x)^3 ) = -G(-x),
%F (4) A( A(x-2*x^3) + 2*A(x-2*x^3)^3 ) = x,
%F where G(x) = x + 2*G(x)^3 = Sum_{n>=1} 2^(n-1) * binomial(3*n-2,n-1)/(3*n-2) * x^(2*n-1).
%e G.f.: A(x) = x + 6*x^5 + 230*x^9 + 11676*x^13 + 663174*x^17 + 41413812*x^21 + 2680851420*x^25 + 188202575736*x^29 + 11594488853190*x^33 +...
%e such that Series_Reversion( A(x) + 2*A(x)^3 ) = A(x) - 2*A(x)^3, where
%e A(x)^3 = x^3 + 18*x^7 + 798*x^11 + 43524*x^15 + 2593398*x^19 + 166441788*x^23 + 11038085612*x^27 + 778999431816*x^31 + 50589526452390*x^35 +...
%e A(x) + 2*A(x)^3 = x + 2*x^3 + 6*x^5 + 36*x^7 + 230*x^9 + 1596*x^11 + 11676*x^13 + 87048*x^15 + 663174*x^17 + 5186796*x^19 + 41413812*x^21 + 332883576*x^23 + 2680851420*x^25 + 22076171224*x^27 + 188202575736*x^29 + 1557998863632*x^31 + 11594488853190*x^33 + 101179052904780*x^35 +...
%e Also,
%e A( A(x) + 2*A(x)^3 ) = x + 2*x^3 + 12*x^5 + 96*x^7 + 880*x^9 + 8736*x^11 + 91392*x^13 +...+ 2^(n-1)*binomial(3*n-2,n-1)/(3*n-2)*x^(2*n-1) +...
%o (PARI) {a(n) = my(Oxn=x*O(x^(4*n)), A = x +Oxn); for(i=1, 4*n, A = A + (x - subst(A + 2*A^3, x, A - 2*A^3 ))/2); polcoeff(A, 4*n-3)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A277292, A277294, A001764.
%K sign
%O 1,2
%A _Paul D. Hanna_, Oct 12 2016
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