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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + x^3 + 7*x^4 + 211*x^5 + 8411*x^6 + 412301*x^7 + 24894581*x^8 + 1832290133*x^9 + 162840289853*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) / (x*A(x)^n)' begins:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, -1, 0, 0, -24, -972, -47184, -2729850, -190252260, ...];
n=2: [1, 0, 1, 0, -35, -1536, -78051, -4655400, -331711815, ...];
n=3: [1, 3, 9, 17, 0, -1674, -94734, -5917068, -433817613, ...];
n=4: [1, 8, 42, 160, 497, 0, -90536, -6434272, -496083426, ...];
n=5: [1, 15, 130, 810, 4075, 16929, 0, -5638950, -504633465, ...];
n=6: [1, 24, 315, 2920, 21396, 132264, 707500, 0, -412691760, ...];
n=7: [1, 35, 651, 8435, 85225, 716457, 5290089, 35515563, 0, ...]; ...
illustrating that the coefficient of x^(n+1) in A(x)^(n^2) / (x*A(x)^n)' equals 0 for n>=0.
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