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EXAMPLE
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G.f.: A(x) = 1 + 5*x + 85*x^2 + 2413*x^3 + 92501*x^4 + 4394663*x^5 + 246960721*x^6 + 15952488893*x^7 + 1161947365721*x^8 + 94123508334877*x^9 + ...
such that
1 = 1/2 + ((1+x) - 3*x*A(x))/2^2 + ((1+x)^2 - 3*x*A(x))^2/2^3 + ((1+x)^3 - 3*x*A(x))^3/2^4 + ((1+x)^4 - 3*x*A(x))^4/2^5 + ((1+x)^5 - 3*x*A(x))^5/2^6 + ...
Also,
1 = 1/(2 + 3*x*A(x)) + (1+x)/(2 + 3*x*A(x)*(1+x))^2 + (1+x)^4/(2 + 3*x*A(x)*(1+x)^2)^3 + (1+x)^9/(2 + 3*x*A(x)*(1+x)^3)^4 + (1+x)^16/(2 + 3*x*A(x)*(1+x)^4)^5 + ...
RELATED SERIES.
Let R(k,x) = Sum_{n>=0} binomial(n+k,k) * (1+x)^(n*(n+k)) / 2^(n+k+1)
then
1 = R(0,x) - 3*x*A(x)*R(1,x) + 3^2*x^2*A(x)^2*R(2,x) - 3^3*x^3*A(x)^3*R(3,x) + 3^4*x^4*A(x)^4*R(4,x) - 3^5*x^5*A(x)^5*R(5,x) + ...
The table of coefficients in R(k,x) begins:
k=0: [1, 3, 36, 744, 21606, 807912, 36948912, 1997801520, ...];
k=1: [1, 10, 197, 5600, 206880, 9387864, 504836996, 31376330400, ...];
k=2: [1, 21, 621, 23447, 1078980, 58590504, 3667676768, ...];
k=3: [1, 36, 1494, 72516, 4075569, 261336096, 18861815280, ...];
k=4: [1, 55, 3050, 185190, 12492745, 934629539, 77091424200, ...];
k=5: [1, 78, 5571, 413764, 33004131, 2850142590, 266518090901, ...];
k=6: [1, 105, 9387, 837165, 77946645, 7696470411, 810015165897, ...];
k=7: [1, 136, 14876, 1568632, 168591350, 18874524760, 2221139481932, ...]; ...
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MATHEMATICA
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nmax = 19;
nmax2 = 300 (* = empirical sum terms *);
sol = {a[0] -> 1};
A[x_] = Sum[a[n] x^n, {n, 0, nmax}];
Do[A[x] = A[x] /. sol; s = 1-Sum[((1+x)^n - 3x A[x] + O[x]^(k+1))^n / 2^(n+1), {n, 0, nmax2}] /. sol; c = SeriesCoefficient[s, {x, 0, k}]; sol = sol ~Join~ Solve[c == 0][[1]] /. HoldPattern[a[n_] -> an_] :> (a[n] -> Round[an]), {k, 2, nmax+1}];
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