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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 25*x^3 + 369*x^4 + 7881*x^5 + 220845*x^6 + 7677363*x^7 + 319307665*x^8 + 15487290535*x^9 + 859400072837*x^10 + ...
such that
A(x) = 1 + x/A(x) + 4*(x/A(x))^2 + 35*(x/A(x))^3 + 506*(x/A(x))^4 + 10472*(x/A(x))^5 + 285384*(x/A(x))^6 +...+ binomial((n+1)^2,n)/(n+1)^2*(x/A(x))^n + ...
RELATED SERIES.
Define B(x) = A(x*B(x)) and A(x) = B(x/A(x)) then B(x) begins
B(x) = 1 + x + 4*x^2 + 35*x^3 + 506*x^4 + 7881*x^5 + 220845*x^6 + 7677363*x^7 + 319307665*x^8 + 15487290535*x^9 + ... + binomial((n+1)^2,n)/(n+1)^2*x^n + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
[1, 1, 3, 25, 369, 7881, 220845, 7677363, 319307665, ...];
[1, 2, 7, 56, 797, 16650, 460291, 15862152, 655825337, ...];
[1, 3, 12, 94, 1293, 26409, 719922, 24587202, 1010428347, ...];
[1, 4, 18, 140, 1867, 37272, 1001476, 33887832, 1384043365, ...];
[1, 5, 25, 195, 2530, 49366, 1306860, 43802060, 1777652015, ...];
[1, 6, 33, 260, 3294, 62832, 1638166, 54370836, 2192294775, ...];
[1, 7, 42, 336, 4172, 77826, 1997688, 65638294, 2629075183, ...];
[1, 8, 52, 424, 5178, 94520, 2387940, 77652024, 3089164371, ...];
[1, 9, 63, 525, 6327, 113103, 2811675, 90463365, 3573805950, ...]; ...
in which the main diagonal begins:
[1, 2, 12, 140, 2530, 62832, 1997688, ..., binomial((n+1)^2,n)/(n+1), ...].
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