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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2233*x^4/4! + 141201*x^5/5! + 12458731*x^6/6! + 1435102663*x^7/7! + 206465053425*x^8/8! + 35963535971233*x^9/9! + ...
such that [x^n] A(x)^(6*n) = (n+5) * [x^(n-1)] A(x)^(6*n) for n>=1.
RELATED SERIES.
A(x)^6 = 1 + 6*x + 48*x^2/2! + 720*x^3/3! + 23328*x^4/4! + 1325376*x^5/5! + 109921536*x^6/6! + 12138398208*x^7/7! + 1692740643840*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(6*n) begins:
n=1: [(1), (6), 24, 120, 972, 55224/5, 763344/5, ...];
n=2: [1, (12), (84), 528, 3960, 197568/5, 2494656/5, ...];
n=3: [1, 18, (180), (1440), 11556, 543672/5, 6306336/5, ...];
n=4: [1, 24, 312, (3072), (27648), 1313856/5, 14451264/5, ...];
n=5: [1, 30, 480, 5640, (57420), (574200), 6220080, ...];
n=6: [1, 36, 684, 9360, 107352, (5759424/5), (63353664/5), ...]; ...
in which the coefficients in parenthesis are related by
6 = 6*(1); 84 = 7*(12); 1440 = 8*(180); 27648 = 9*(3072); 574200 = 10*(57420); 63353664/5 = 11*(5759424/5); ...
illustrating: [x^n] A(x)^(6*n) = (n + 5) * [x^(n-1)] A(x)^(6*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 5*x*A'(x)/A(x)) / (1 - 6*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 8*x^3 + 84*x^4 + 1080*x^5 + 16056*x^6 + 266256*x^7 + 4816080*x^8 + 93638016*x^9 + 1937252160*x^10 + ... + A300993(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)^(6*(#A-1))); A[#A] = ((#A+4)*V[#A-1] - V[#A])/(6*(#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-5*x*A')/(A-6*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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