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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 49*x^3/3! + 1777*x^4/4! + 101541*x^5/5! + 8140411*x^6/6! + 855134533*x^7/7! + 112545136929*x^8/8! + 17984228218057*x^9/9! + ...
such that [x^n] A(x)^(5*n) = (n+4) * [x^(n-1)] A(x)^(5*n) for n>=1.
RELATED SERIES.
A(x)^5 = 1 + 5*x + 35*x^2/2! + 485*x^3/3! + 14545*x^4/4! + 756025*x^5/5! + 57290875*x^6/6! + 5790439625*x^7/7! + 740641270625*x^8/8! + 115751765142125*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(5*n) begins:
n=1: [(1), (5), 35/2, 485/6, 14545/24, 151205/24, ...];
n=2: [1, (10), (60), 1010/3, 6980/3, 21490, 2249000/9, ...];
n=3: [1, 15, (255/2), (1785/2), 51795/8, 449805/8, ...];
n=4: [1, 20, 220, (5620/3), (44960/3), 389740/3, ...];
n=5: [1, 25, 675/2, 20425/6, (730225/24), (2190675/8), ...];
n=6: [1, 30, 480, 5610, 55980, (534270), (5342700), ...]; ...
in which the coefficients in parenthesis are related by
5 = 5*(1); 60 = 6*(10); 1785/2 = 7*(255/2); 44960/3 = 8*(5620/3); 2190675/8 = 9*(730225/24); 5342700 = 10*(534270); ...
illustrating: [x^n] A(x)^(5*n) = (n+4) * [x^(n-1)] A(x)^(5*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 4*x*A'(x)/A(x)) / (1 - 5*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 7*x^3 + 66*x^4 + 769*x^5 + 10405*x^6 + 157540*x^7 + 2609120*x^8 + 46569365*x^9 + 886686635*x^10 + ... + A300991(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)^(5*(#A-1))); A[#A] = ((#A+3)*V[#A-1] - V[#A])/(5*(#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-4*x*A')/(A-5*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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