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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 32*x^3 + 813*x^4 + 35091*x^5 + 2239211*x^6 + 196853182*x^7 + 22793279166*x^8 + ...
We will use the following tables to illustrate the definition of the g.f. A(x).
The table of coefficients of x^k in A(x)^n begins:
n=1: [1, 1, 3, 32, 813, 35091, 2239211, 196853182, ...];
n=2: [1, 2, 7, 70, 1699, 72000, 4554506, 398447364, ...];
n=3: [1, 3, 12, 115, 2667, 110850, 6948927, 604906389, ...];
n=4: [1, 4, 18, 168, 3727, 151776, 9425698, 816358612, ...];
n=5: [1, 5, 25, 230, 4890, 194926, 11988240, 1032937150, ...];
n=6: [1, 6, 33, 302, 6168, 240462, 14640187, 1254780150, ...];
n=7: [1, 7, 42, 385, 7574, 288561, 17385403, 1482031076, ...];
...
and the table of coefficients of x^k in 1/A(x)^n starts:
n=1: [1,-1, -2, -27, -748, -33385, -2166001, -192231746, ...];
n=2: [1,-2, -3, -50,-1438, -65166, -4261511, -379957558, ...];
n=3: [1,-3, -3, -70,-2076, -95436, -6289106, -563289078, ...];
n=4: [1,-4, -2, -88,-2667,-124280, -8251230, -742334276, ...];
n=5: [1,-5, 0,-105,-3215,-151776,-10150205, -917197625, ...];
n=6: [1,-6, 3,-122,-3723,-177996,-11988240,-1087980264, ...];
n=7: [1,-7, 7,-140,-4193,-203007,-13767439,-1254780150, ...];
...
now, upon comparing these two tables, we see that
[x^n] A(x)^(n-1) = [x^n] -1/A(x)^n for n > 1,
as illustrated by
[x^2] A(x) = 3 = [x^2] -1/A(x)^2 = -(-3);
[x^3] A(x)^2 = 70 = [x^3] -1/A(x)^3 = -(-70);
[x^4] A(x)^3 = 2667 = [x^4] -1/A(x)^4 = -(-2667);
[x^5] A(x)^4 = 151776 = [x^5] -1/A(x)^5 = -(-151776);
[x^6] A(x)^5 = 11988240 = [x^6] -1/A(x)^6 = -(-11988240);
[x^7] A(x)^6 = 1254780150 = [x^7] -1/A(x)^7 = -(-1254780150);
...
Incidentally, note that the coefficients along the diagonals in the above tables form the series
log(B(x)) = x + 3*x^2/2 + 70*x^3/3 + 2667*x^4/4 + 151776*x^5/5 + 11988240*x^6/6 + 1254780150*x^7/7 + ...
where
B(x) = A(x/B(x)) = 1 + x + 2*x^2 + 25*x^3 + 692*x^4 + 31070*x^5 + 2030041*x^6 + 181330044*x^7 + ...
Further, let
C(x) = A(x*C(x)) = 1 + x + 4*x^2 + 42*x^3 + 978*x^4 + 40077*x^5 + 2483629*x^6 + 214354877*x^7 + ...
then
(C(x) + x*C'(x))/C(x)^2 - (1-x) = x*B'(x)/B(x) = x + 3*x^2 + 70*x^3 + 2667*x^4 + 151776*x^5 + 11988240*x^6 + ...
which is due to the property of g.f. A(x) given in the definition.
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