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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 54*x^2 + 4410*x^3 + 2208465*x^4 +...
A(x) = 1 - log(eta(3*x)) + log(eta(9*x))^2/2! - log(eta(27*x))^3/3! +-...
...
Let P(x) = 1/eta(x) denote the g.f. of the partition numbers A000041:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 +...
then a(n) is the coefficient of x^n in P(x)^(3^n):
P(x)^(3^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
P(x)^(3^1): [1,(3),9,22,51,108,221,429,810,1479,2640,4599,...];
P(x)^(3^2): [1,9,(54),255,1035,3753,12483,38709,113265,...];
P(x)^(3^3): [1,27,405,(4410),38745,290466,1923075,11506185,...];
P(x)^(3^4): [1,81,3402,98523,(2208465),40795083,645824907,...];
P(x)^(3^5): [1,243,29889,2480382,156189951,(7958364696),...];
where terms in parenthesis form the initial terms of this sequence.
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PROG
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(PARI) {a(n)=polcoeff(1/eta(x+x*O(x^n))^(3^n), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, (-1)^m*log(eta(3^m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, (3^m*x)^k/(1-(3^m*x)^k)/k+x*O(x^n))^m/m!), n)}
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