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EXAMPLE
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G.f.: A(x) = 1 - 5*x + 275*x^2 - 302250*x^3 + 6175682500*x^4 +...
A(x) = 1 + log(eta(5*x)) + log(eta(25*x))^2/2! + log(eta(125*x))^3/3! +...
...
Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...
then a(n) is the coefficient of x^n in eta(x)^(5^n):
eta(x)^(5^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,..];
eta(x)^(5^1): [1,(-5),5,10,-15,-6,-5,25,15,-20,9,-45,-5,25,...];
eta(x)^(5^2): [1,-25,(275),-1700,6050,-9405,-15550,107525,...];
eta(x)^(5^3): [1,-125,7625,(-302250),8745875,-196718900,...];
eta(x)^(5^4): [1,-625,194375,-40105000,(6175682500),...];
where terms in parenthesis form the initial terms of this sequence.
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PROG
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(PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(5^n), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, log(eta(5^m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, -(5^m*x)^k/(1-(5^m*x)^k)/k+x*O(x^n))^m/m!), n)}
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