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EXAMPLE
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E.g.f.: A(x) = 1 + x + 5*x^2/2! + 445*x^3/3! + 196105*x^4/4! + 221673401*x^5/5! + 501981700621*x^6/6! + 1983064113021685*x^7/7! + 12488526496641458705*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^3*x) / A(x)^(n^2) begins:
n=1: [1, 0, -4, -432, -194256, -220662720, -500627544000, ...];
n=2: [1, 4, 0, -1856, -805376, -898258176, -2023715201024, ...];
n=3: [1, 18, 288, 0, -1989792, -2154563712, -4727980751616, ...];
n=4: [1, 48, 2240, 94464, 0, -4244861952, -9137589559296, ...];
n=5: [1, 100, 9900, 959200, 84852400, 0, -15901448888000, ...];
n=6: [1, 180, 32256, 5738688, 1003636224, 161358324480, 0, ...];
n=7: [1, 294, 86240, 25218144, 7335234144, 2103824749824, 557359956846336, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 2*x^2 + 72*x^3 + 8096*x^4 + 1839000*x^5 + 695334816*x^6 + 392764566208*x^7 + 309340607492096*x^8 + ... + A317346(n)*x^n + ...
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