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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1945*x^4/4! + 178041*x^5/5! + 26792971*x^6/6! + 5940440563*x^7/7! + 1812303908913*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^2*x) / A(x)^n begins:
n=1: [1, 0, -2, -36, -1764, -167280, -25620600, -5737974480, ...];
n=2: [1, 2, 0, -88, -4160, -371328, -55329536, -12201990400, ...];
n=3: [1, 6, 30, 0, -7812, -698184, -97733304, -20791334880, ...];
n=4: [1, 12, 136, 1296, 0, -1171968, -168658176, -33909447168, ...];
n=5: [1, 20, 390, 7220, 113020, 0, -265712600, -55963975600, ...];
n=6: [1, 30, 888, 25704, 709056, 16600320, 0, -84622337280, ...];
n=7: [1, 42, 1750, 72072, 2909340, 112245672, 3684715944, 0, ...];
n=8: [1, 56, 3120, 172640, 9455488, 508540416, 26199517696, 1150524892160, 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 + x^2 + 6*x^3 + 74*x^4 + 1400*x^5 + 35676*x^6 + 1140328*x^7 + 43740848*x^8 + 1954336608*x^9 + 99561874080*x^10 + ... + A317344(n)*x^n + ...
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