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EXAMPLE
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E.g.f.: A(x) = 1 + x + 9*x^2/2! + 1483*x^3/3! + 976825*x^4/4! + 1507281021*x^5/5! + 4409747597401*x^6/6! + 21744850191313999*x^7/7! + 167557834535988306033*x^8/8! + 1913194223179191462419065*x^9/9! + 31110747474489521617502800201*x^10/10! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^3) begins:
n=1: [(1), (1), 9/2, 1483/6, 976825/24, 502427007/40, 4409747597401/720]
n=2: [1, (8), (64), 6856/3, 1022528/3, 1543097816/15, 2237393526784/45]
n=3: [1, 27, (945/2), (25515/2), 10692675/8, 14849374869/40, 1397853444500
n=4: [1, 64, 2304, (226880/3), (14520320/3), 5124803136/5, 20241220116736/
n=5: [1, 125, 16625/2, 2510375/6, (553359625/24), (69169953125/24), ...];
n=6: [1, 216, 24192, 1918728, 131302080, (56555402904/5), (12215967027264/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 64 = 2^3*8; 25515/2 = 3^3*945/2; 14520320/3 = 4^3*226880/3; ...
illustrating that: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + 4308276119854*x^7 + 4151360558858752*x^8 + 5268077625693186225*x^9 + 8567999843251994553500*x^10 + ... + A300595(n)*x^n + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1)^3 ); EGF=Ser(A); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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