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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 126*x^3/3! + 1928*x^4/4! + 39050*x^5/5! +...
The exponential of the e.g.f. begins:
exp(x*A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 721*x^4/4! + 14241*x^5/5! +...
The coefficients of x^n/n! in the powers of G(x) = 1 + 2*x*exp(x) begin:
G^1: [(1), 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...];
G^2: [1,(4), 16, 60, 208, 660, 1944, 5404, 14368, 36900 ...];
G^3: [1, 6,(36), 210, 1176, 6270, 31716, 152250, 696240, ...];
G^4: [1, 8, 64, (504), 3872, 28840, 207408, 1436792, ...];
G^5: [1, 10, 100, 990, (9640), 91890, 854460, 7731430, ...];
G^6: [1, 12, 144, 1716, 20208,(234300), 2666952, 29736084, ...];
G^7: [1, 14, 196, 2730, 37688, 514150, (6914964), 91510034, ...];
G^8: [1, 16, 256, 4080, 64576, 1012560, 15698016,(240229360), ...]; ...
where the coefficients in parenthesis form initial terms of this sequence:
[1/1, 4/2, 36/3, 504/4, 9640/5, 234300/6, 6914964/7, 240229360/8, ...].
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