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A341842
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Coefficients of the series whose 12th power equals E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009.
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0
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1, 18, -2088, 301296, -50784174, 9174627360, -1734603719472, 338286925650240, -67486440186470016, 13697820033167444178, -2818359890320927630320, 586296297186462310481424, -123077156275866375661524864, 26034142700316716015964656544
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OFFSET
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0,2
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COMMENTS
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The g.f. is the 12th root of the g.f. of A282019.
It is easy to see that E_2(x)*E_4(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 is always divisible by 3. Hence, E_2(x)*E_4(x) == 1 (mod 72). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x))^(1/12) = 1 + 18*x - 2088*x^2 + 301296*x^3 - 50784174*x^4 + ... has integer coefficients.
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LINKS
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MAPLE
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E(2, x) := 1 - 24*add(k*x^k/(1-x^k), k = 1..20):
E(4, x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2, x)*E(4, x))^(1/12), x, 20):
seriestolist(%);
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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