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EXAMPLE
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E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 278*x^3/3! + 5128*x^4/4! + 125592*x^5/5! + 3850000*x^6/6! + 142013328*x^7/7! + ...
such that A(x) = exp( 2*x*A(x) ) / (1-x), where
exp( 2*x*A(x) ) = 1 + 2*x + 16*x^2/2! + 212*x^3/3! + 4016*x^4/4! + 99952*x^5/5! + 3096448*x^6/6! + 115063328*x^7/7! + ...
Related table.
Another interesting property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in 1/A(x)^n begins:
n=1: [1, -3, -4, -44, -736, -16832, -491168, ...];
n=2: [1, -6, 10, -16, -320, -8064, -249344, ...];
n=3: [1, -9, 42, -78, -48, -1776, -66528, ...];
n=4: [1, -12, 92, -392, 728, -128, -8960, ...];
n=5: [1, -15, 160, -1120, 4600, -8520, -320, ...];
n=6: [1, -18, 246, -2424, 16104, -64752, 119952, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1, as follows:
n=1:-2 = 1 + -3;
n=2: 0 = 1 + -6 + 10/2!;
n=3: 0 = 1 + -9 + 42/2! + -78/3!;
n=4: 0 = 1 + -12 + 92/2! + -392/3! + 728/4!;
n=5: 0 = 1 + -15 + 160/2! + -1120/3! + 4600/4! + -8520/5!;
n=6: 0 = 1 + -18 + 246/2! + -2424/3! + 16104/4! + -64752/5! + 119952/6!;
...
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