Raising to the power x, we have S = (1+S)^x.
Letting x -> 0, we conclude that S = 1 + x*T for some power series x. Taking logarithm, we then have
log(1 + x*T) = x*log(2 + x*T) = x*log(2) + x*log(1+x*T/2)
That is, we have the power series identity
x*log(2) + Sum_{k>=1} (-1)^(k-1) * x^k * T^k * (x - 2^k) / k / 2^k = 0.
Let a_m denote the coefficient of x^m in T.
Note that in the coefficient of x^(m+1) in the last identity, a_m comes in the first degree and with the coefficient -1. This allows to determine the coefficients a_m iteratively as illustrated by the following code (where L stands for log(2)):
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? T = sum(i=0,5, eval(concat("a",Str(i)))*x^i) + O(x^6)
%1 = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 + a5*x^5 + O(x^6)
? EQ = x*L + sum(k=1,6, (-1)^(k-1) * x^k * T^k * (x-2^k) / k / 2^k)
%2 = (-a0 + L)*x + (1/2*a0^2 + 1/2*a0 - a1)*x^2 + (-1/3*a0^3 - 1/8*a0^2 + a1*a0 + (1/2*a1 - a2))*x^3 + (1/4*a0^4 + 1/24*a0^3 - a1*a0^2 + (-1/4*a1 + a2)*a0 + (1/2*a1^2 + (1/2*a2 - a3)))*x^4 + (-1/5*a0^5 - 1/64*a0^4 + a1*a0^3 + (1/8*a1 - a2)*a0^2 + (-a1^2 + (-1/4*a2 + a3))*a0 + (-1/8*a1^2 + a2*a1 + (1/2*a3 - a4)))*x^5 + (1/6*a0^6 + 1/160*a0^5 - a1*a0^4 + (-1/16*a1 + a2)*a0^3 + (3/2*a1^2 + (1/8*a2 - a3))*a0^2 + (1/8*a1^2 - 2*a2*a1 + (-1/4*a3 + a4))*a0 + (-1/3*a1^3 + (-1/4*a2 + a3)*a1 + (1/2*a2^2 + (1/2*a4 - a5))))*x^6 + O(x^7)
? a=vector(6); for(i=0,5, c = polcoef(EQ,i+1); for(j=0,i-1, c=subst(c,variable(polcoef(T,j)),a[j+1])); a[i+1] = polcoef(c,0)); a
%3 = [L, 1/2*L^2 + 1/2*L, 1/6*L^3 + 5/8*L^2 + 1/4*L, 1/24*L^4 + 3/8*L^3 + 9/16*L^2 + 1/8*L, 1/120*L^5 + 9/64*L^4 + 17/32*L^3 + 7/16*L^2 + 1/16*L, 1/720*L^6 + 7/192*L^5 + 55/192*L^4 + 115/192*L^3 + 5/16*L^2 + 1/32*L]
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Regards,
Max