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EXAMPLE
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O.g.f.: A(x) = 1 + x + 37*x^2 + 4096*x^3 + 878619*x^4 + 306873869*x^5 + 158938884952*x^6 + 114993958088544*x^7 + 111352808890827351*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x)^3 dx)/A(x) begins:
n=0: [1, -1, -36, -4023, -869168, -304829775, ...];
n=1: [1, 0, -35, -12064/3, -870135, -915526348/3, ...];
n=2: [1, 7, 0, -11609/3, -2626022/3, -307526817, ...];
n=3: [1, 26, 342, 0, -847892, -312911550, ...];
n=4: [1, 63, 2044, 131387/3, 0, -919948381/3, ...];
n=5: [1, 124, 7839, 1011556/3, 31877746/3, 0], ...];
n=6: [1, 215, 23400, 1722357, 96411130, 4177156347, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x)^3 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 75*x^2 + 8266*x^3 + 1766799*x^4 + 615808080*x^5 + 318573312664*x^6 + 230335700260146*x^7 + 222950653057400247*x^8 + ...
A(x)^3 = 1 + 3*x + 114*x^2 + 12511*x^3 + 2664651*x^4 + 926819028*x^5 + 478906878958*x^6 + 346026409343751*x^7 + 334794104506072215*x^8 + ...
exp( Integral A(x)^3 dx) = 1 + x + 7*x^2/2! + 703*x^3/3! + 303145*x^4/4! + 321307921*x^5/5! + 669264720031*x^6/6! + 2418416266536607*x^7/7! + 13971240948079459633*x^8/8! + ...
A'(x)/A(x) = 1 + 73*x + 12178*x^2 + 3495501*x^3 + 1529245631*x^4 + 951553836400*x^5 + 803743212623394*x^6 + 889843851811684197*x^7 + ...
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