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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 38*x^3 + 931*x^4 + 32226*x^5 + 1415534*x^6 + 74544428*x^7 + 4548135075*x^8 + 314358016202*x^9 + ...
such that [x^n] A(x)^(2*n+1) / (x*A(x)^2)' = 0 for n>1.
ILLUSTRATION OF DEFINITION.
The table of coefficients in A(x)^(2*n+1) / (x*A(x)^2)' begins:
n=0: [1, -3, -6, -203, -6882, -304062, -16218796, -1004084811, ...];
n=1: [1, -1, -5, -154, -5629, -259038, -14200498, -896533268, ...];
n=2: [1, 1, 0, -89, -4107, -207189, -11944332, -778560170, ...];
n=3: [1, 3, 9, 0, -2256, -147537, -9425665, -649322628, ...];
n=4: [1, 5, 22, 121, 0, -78928, -6616832, -507901145, ...];
n=5: [1, 7, 39, 282, 2753, 0, -3486672, -353291008, ...];
n=6: [1, 9, 60, 491, 6111, 90849, 0, -184392528, ...];
n=7: [1, 11, 85, 756, 10198, 195493, 3882985, 0, ...]; ...
in which the main diagonal consists of all zeros after the initial terms, illustrating that [x^n] A(x)^(2*n+1) / (x*A(x)^2)' = 0 for n>1.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 7*x^2 + 82*x^3 + 1947*x^4 + 66542*x^5 + 2902550*x^6 + 152184036*x^7 + 9257168147*x^8 + ...
(x*A(x)^2)' = 1 + 4*x + 21*x^2 + 328*x^3 + 9735*x^4 + 399252*x^5 + 20317850*x^6 + 1217472288*x^7 + 83314513323*x^8 + ...
log(A(x)) = x + 5*x^2/2 + 106*x^3/3 + 3565*x^4/4 + 156126*x^5/5 + 8285474*x^6/6 + 511246324*x^7/7 + 35754687997*x^8/8 + 2786287728022*x^9/9 + ...
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