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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1009*x^4/4! + 44541*x^5/5! + 2799931*x^6/6! + 233188033*x^7/7! + 24562692897*x^8/8! + 3168510747769*x^9/9! + 488856473079571*x^10/10! + ...
such that [x^n] A(x)^(3*n) = (n+2) * [x^(n-1)] A(x)^(3*n) for n>=1.
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2/2! + 171*x^3/3! + 4185*x^4/4! + 173583*x^5/5! + 10491039*x^6/6! + 850141575*x^7/7! + 87745941873*x^8/8! + 11141030530395*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in A(x)^(3*n) begins:
n=1: [(1), (3), 15/2, 57/2, 1395/8, 57861/40, 1165671/80, 18892035/112, ...];
n=2: [1, (6), (24), 102, 576, 21834/5, 206244/5, 15974712/35, ...];
n=3: [1, 9, (99/2), (495/2), 11259/8, 401463/40, 7120899/80, 525246849/560, ...];
n=4: [1, 12, 84, (492), (2952), 102708/5, 864756/5, 60722784/35, ...];
n=5: [1, 15, 255/2, 1725/2, (44595/8), (312165/8), 5077035/16, 340795215/112, ...];
n=6: [1, 18, 180, 1386, 9720, (349542/5), (2796336/5), 36178488/7, ...];
n=7: [1, 21, 483/2, 4179/2, 127323/8, 4767147/40, (76271139/80), (686440251/80), ...]; ...
in which the coefficients in parenthesis are related by
3 = 3*(1); 24 = 4*(6); 495/2 = 5*(99/2); 2952 = 6*(492); 312165/8 = 7*(44595/8); 2796336/5 = 8*(349542/5); 686440251/80 = 9*(76271139/80); ...
illustrating that: [x^n] A(x)^(3*n) = (n+2) * [x^(n-1)] A(x)^(3*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 2*x*A'(x)/A(x)) / (1 - 3*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 5*x^3 + 36*x^4 + 327*x^5 + 3489*x^6 + 42048*x^7 + 559008*x^8 + 8073243*x^9 + 125328411*x^10 + 2075525505*x^11 + 36460943208*x^12 + ... + A300987(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)^(3*(#A-1))); A[#A] = ((#A+1)*V[#A-1] - V[#A])/(3*(#A-1)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = exp( x*(A-2*x*A')/(A-3*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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