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A341871
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Coefficients of the series whose 48th power equals E_2(x)^2/E_4(x), where E_2(x) and E_4(x) are the Eisenstein series A006352 and A004009.
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4
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1, -6, 558, -88884, 15433662, -2864048616, 552921962724, -109731286565040, 22220439670517814, -4569456313225317114, 951159953810624453208, -199945837161334089352548, 42373766861587365894611604
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OFFSET
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0,2
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COMMENTS
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It is easy to see that E_2(x)^2/E_4(x) == 1 - 48*Sum_{k >= 1} (k + 5*k^3)*x^k/(1 - x^k) (mod 288), and also that the integer k + 5*k^3 is always divisible by 6. Hence, E_2(x)^2/E_4(x) == 1 (mod 288). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^2/E_4(x))^(1/48) = 1 - 6*x + 558*x^2 - 88884*x^3 + 15433662*x^4 - ... has integer coefficients.
Note that (E_2(x)^2/E_4(x))^(1/48) = (E_2(x)^4/E_8(x))^(1/96).
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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)^2/E(4, x))^(1/48), 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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