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EXAMPLE
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O.g.f.: A(x) = x + 9*x^2 + 552*x^3 + 85842*x^4 + 24653700*x^5 + 11219022936*x^6 + 7393496092416*x^7 + 6649411839351120*x^8 + ...
such that [x^n] exp( n^3*x - n*A(x) ) = 0 for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients in begins:
n=1: [1, 0, -18, -3312, -2059236, -2957847840, -8077030651800, ...];
n=2: [1, 6, 0, -7056, -4281984, -6040453824, -16367904244224, ...];
n=3: [1, 24, 522, 0, -6980580, -9667325376, -25560523291464, ...];
n=4: [1, 60, 3528, 189792, 0, -14146669440, -37025599219200, ...];
n=5: [1, 120, 14310, 1679040, 181358460, 0, -51097553724600, ...];
n=6: [1, 210, 43992, 9173088, 1887214464, 358972896960, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 19*x^2/2! + 3367*x^3/3! + 2074537*x^4/4! + 2969379361*x^5/5! + 8096147776171*x^6/6! + 37321188279552199*x^7/7! + ...
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PROG
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(PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))
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