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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 152*x^2 + 49920*x^3 + 191701440*x^4 +...
A(x) = 1 - log(eta(4*x)) + log(eta(16*x))^2/2! - log(eta(64*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)^(4^n):
P(x)^(4^0): [(1),1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,...];
P(x)^(4^1): [1,(4),14,40,105,252,574,1240,2580,5180,10108,...];
P(x)^(4^2): [1,16,(152),1088,6460,33440,155584,663936,2636326,...];
P(x)^(4^3): [1,64,2144,(49920),905840,13627264,176638592,...];
P(x)^(4^4): [1,256,33152,2894848,(191701440),10266643968,...];
P(x)^(4^5): [1,1024,525824,180531200,46620870400,(9659304851456),...];
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))^(4^n), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, (-1)^m*log(eta(4^m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, (4^m*x)^k/(1-(4^m*x)^k)/k+x*O(x^n))^m/m!), n)}
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