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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 697*x^4/4! + 25761*x^5/5! + 1371691*x^6/6! + 97677343*x^7/7! + 8869533681*x^8/8! + 993709302337*x^9/9! + 134086553693011*x^10/10! + ...
such that [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n) for n>=1.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1696*x^4/4! + 60352*x^5/5! + 3134464*x^6/6! + 219316736*x^7/7! + 19655797760*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(2*n) begin:
n=1: [(1), (2), 4, 40/3, 212/3, 7544/15, 195904/45, 13707296/315, ...];
n=2: [1, (4), (12), 128/3, 632/3, 6976/5, 515776/45, 34760896/315, ...];
n=3: [1, 6, (24), (96), 468, 14664/5, 114384/5, 7407552/35, ...];
n=4: [1, 8, 40, (544/3), (2720/3), 82496/15, 1843264/45, 22923136/63, ...];
n=5: [1, 10, 60, 920/3, (4820/3), (9640), 622880/9, 37242080/63, ...];
n=6: [1, 12, 84, 480, 2664, (80448/5), (563136/5), 32495424/35, ...];
n=7: [1, 14, 112, 2128/3, 12572/3, 387128/15, (8018416/45), (64147328/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*(1); 12 = 3*(4); 96 = 4*(24); 2720/3 = 5*(544/3); 9640 = 6*(4820/3); 563136/5 = 7*(80448/5); 64147328/45 = 8*(8018416/45); ...
illustrating that: [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - x*A'(x)/A(x)) / (1 - 2*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 4*x^3 + 24*x^4 + 184*x^5 + 1672*x^6 + 17296*x^7 + 198800*x^8 + 2499200*x^9 + 33992000*x^10 + 496281344*x^11 + 7731823616*x^12 + ...
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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)^(2*(#A-1))); A[#A] = ((#A)*V[#A-1] - V[#A])/(2*(#A-1)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = exp( x*(A-x*A')/(A-2*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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