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EXAMPLE
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E.g.f. A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4/4! +...
A(x/A(x)) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...
1/A(x) = 1 + x - x^2/2! + 10*x^3/3! - 159*x^4/4! + 3816*x^5/5! -+...
A(log(A(x)) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^n for n>=1, k>=0:
exp(x)*A(x)^1: [(1), 2, 6, 35, 416, 8437, 249340, ...];
exp(x)*A(x)^2: [1,(3), 13, 93, 1145, 22593, 645741, ...];
exp(x)*A(x)^3: [1, 4,(22), 181, 2320, 45199, 1257364, ...];
exp(x)*A(x)^4: [1, 5, 33,(305), 4097, 79825, 2177329, ...];
exp(x)*A(x)^5: [1, 6, 46, 471,(6656), 131001, 3529836, ...];
exp(x)*A(x)^6: [1, 7, 61, 685, 10201,(204337), 5477005, ...];
exp(x)*A(x)^7: [1, 8, 78, 953, 14960, 306643,(8226436), ...]; ...
then the terms along the main diagonal form this sequence shift left.
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