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EXAMPLE
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E.g.f.: A(x) = 1 + x + 5*x^2/2! + 49*x^3/3! + 745*x^4/4! + 16001*x^5/5! + 472621*x^6/6! + 19659025*x^7/7! + 1211940689*x^8/8! + 112956505345*x^9/9! + ...
The table of coefficients in A(x)^(-n) begins:
n=1: [(1), (-1), -3/2, -25/6, -359/24, -2587/40, -245099/720, ...];
n=2: [1, (-2), (-2), -16/3, -58/3, -1304/15, -22016/45, ...];
n=3: [1, -3, (-3/2), (-9/2), -141/8, -3441/40, -42579/80, ...];
n=4: [1, -4, 0, (-8/3), (-40/3), -376/5, -23624/45, -293096/63, ...];
n=5: [1, -5, 5/2, -5/6, (-215/24), (-1505/24), -72055/144, ...];
n=6: [1, -6, 6, 0, -6, (-264/5), (-2376/5), -173388/35, ...];
n=7: [1, -7, 21/2, -7/6, -119/24, -1869/40, (-327971/720), (-3607681/720), ...]; ...
in which the coefficients in parenthesis are related by
-1 = -1*(1); -2 = 1*(-2); -9/2 = 3*(3/2); -40/3 = 5*(-8/3); -1505/24 = 7*(-215/24); -2376/5 = 9*(-264/5); -3607681/720 = 11*(-327971/720); ...
illustrating: [x^n] A(x)^(-n) = (2*n - 3) * [x^(n-1)] A(x)^(-n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 + 3*x*A'(x)/A(x)) / (1 + x*A'(x)/A(x));
explicitly,
log(A(x)) = x + 2*x^2 + 6*x^3 + 22*x^4 + 94*x^5 + 474*x^6 + 2974*x^7 + 24630*x^8 + 271710*x^9 + 3799570*x^10 + ... + A301385(n)*x^n + ...
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