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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 39*x^3 + 1300*x^4 + 68495*x^5 + 5122036*x^6 + 513764594*x^7 + 66569992756*x^8 + ...
where the coefficient of x^n in A(x)^(2*n) equals the coefficient of x^n in the sum A(x)^n + A(x)^(n+1), for n > 1.
To illustrate, we may inspect the table of coefficients of x^k in A(x)^n, which begins:
n=1: [1, 1, 2, 39, 1300, 68495, 5122036, ...];
n=2: [1, 2, 5, 82, 2682, 139746, 10387783, ...];
n=3: [1, 3, 9, 130, 4152, 213882, 15801617, ...];
n=4: [1, 4, 14, 184, 5717, 291040, 21368094, ...];
n=5: [1, 5, 20, 245, 7385, 371366, 27091960, ...];
n=6: [1, 6, 27, 314, 9165, 455016, 32978162, ...];
n=7: [1, 7, 35, 392, 11067, 542157, 39031860, ...];
n=8: [1, 8, 44, 480, 13102, 632968, 45258440, ...];
n=9: [1, 9, 54, 579, 15282, 727641, 51663528, ...];
n=10: [1, 10, 65, 690, 17620, 826382, 58253005, ...];
n=11: [1, 11, 77, 814, 20130, 929412, 65033023, ...];
n=12: [1, 12, 90, 952, 22827, 1036968, 72010022, ...];
...
to see the following pattern:
[x^2] A(x)^2 + A(x)^3 = 5 + 9 = [x^2] A(x)^4 = 14 ;
[x^3] A(x)^3 + A(x)^4 = 130 + 184 = [x^3] A(x)^6 = 314 ;
[x^4] A(x)^4 + A(x)^5 = 5717 + 7385 = [x^4] A(x)^8 = 13102 ;
[x^5] A(x)^5 + A(x)^6 = 371366 + 455016 = [x^5] A(x)^10 = 826382 ;
[x^6] A(x)^6 + A(x)^7 = 32978162 + 39031860 = [x^6] A(x)^12 = 72010022 ;
...
Incidentally, note that these coefficients from the above table form the series
log(B(x)^2) = 2*x + 14*x^2/2 + 314*x^3/3 + 13102*x^4/4 + 826382*x^5/5 + 72010022*x^6/6 + 8248064778*x^7/7 + ...
where
B(x) = A(x*B(x)^2) = 1 + x + 4*x^2 + 56*x^3 + 1698*x^4 + 84492*x^5 + 6091596*x^6 + 595572378*x^7 + 75718532850*x^8 + ...
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