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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 8*x^2 + 84*x^3 + 1522*x^4 + 38940*x^5 + 1278880*x^6 + 51136512*x^7 + 2407921070*x^8 + 130495143060*x^9 + 8002598818976*x^10 +...
such that
A(x) = 1 + 2*(x/A(x)) + 12*(x/A(x))^2 + 140*(x/A(x))^3 + 2530*(x/A(x))^4 + 62832*(x/A(x))^5 + 1997688*(x/A(x))^6 +...+ binomial((n+1)^2,n)/(n+1)*(x/A(x))^n +...
The table of coefficients of x^k in A(x)^(n+1) begins:
[1, 2, 8, 84, 1522, 38940, 1278880, 51136512, 2407921070, ...];
[1, 4, 20, 200, 3444, 85312, 2744928, 108267280, 5049708672, ...];
[1, 6, 36, 356, 5862, 140508, 4424984, 172064160, 7946443350, ...];
[1, 8, 56, 560, 8888, 206176, 6350112, 243284064, 11121338640, ...];
[1, 10, 80, 820, 12650, 284252, 8556240, 322780800, 14599990830, ...];
[1, 12, 108, 1144, 17292, 376992, 11084864, 411518448, 18410660208, ...];
[1, 14, 140, 1540, 22974, 487004, 13983816, 510586400, 22584587382, ...];
[1, 16, 176, 2016, 29872, 617280, 17308096, 621216192, 27156348512, ...];
[1, 18, 216, 2580, 38178, 771228, 21120768, 744800256, 32164253550, ...]; ...
in which the main diagonal begins:
[1, 4, 36, 560, 12650, 376992, 13983816, ..., binomial((n+1)^2,n), ...],
thus [x^n] A(x)^(n+1) = [x^n] (1 + x)^((n+1)^2) for n>=0.
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