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A279929
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Expansion of c(q)*c(q^2)/9 - c(q^3)*c(q^6)/3 in powers of q where c() is a cubic AGM theta function.
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1
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1, 1, 0, 1, 6, 0, 8, 1, 0, 6, 12, 0, 14, 8, 0, 1, 18, 0, 20, 6, 0, 12, 24, 0, 31, 14, 0, 8, 30, 0, 32, 1, 0, 18, 48, 0, 38, 20, 0, 6, 42, 0, 44, 12, 0, 24, 48, 0, 57, 31, 0, 14, 54, 0, 72, 8, 0, 30, 60, 0, 62, 32, 0, 1, 84, 0, 68, 18, 0, 48, 72, 0, 74, 38, 0
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OFFSET
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1,5
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COMMENTS
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LINKS
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FORMULA
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Expansion of (2*a(q)^2 - a(q)*a(q^2) - 4*a(q^2)^2 + 3*a(q^3)*a(q^6)) / 18 in powers of q where a() is a cubic AGM theta function.
Expansion of (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2)) - 3 * (eta(q^9) * eta(q^18))^3 / (eta(q^3) * eta(q^6)) in powers of q.
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 0^e, a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.
G.f. is a period 1 Fourier series that satisfies f(-1 / (18 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A281786.
a(2*n) = a(n). a(3*n) = 0.
Dirichlet g.f.: zeta(s-1) * zeta(s) * (1 - 2^(1-s)) * (1 - 3^(1-s)) * (1 - 3^(-s)).
Sum_{k=1..n} a(k) ~ (2*Pi^2/81) * n^2. (End)
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EXAMPLE
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G.f. = q + q^2 + q^4 + 6*q^5 + 8*q^7 + q^8 + 6*q^10 + 12*q^11 + 14*q^13 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, 1, # == 3, 0, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)];
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)) - 3 * x^2 * (eta(x^9 + A) * eta(x^18 + A))^3 / (eta(x^3 + A) * eta(x^6 + A)), n))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, 0, (p^(e+1) - 1) / (p - 1))))};
(Magma) A := Basis( ModularForms( Gamma0(18), 2), 75); A[2] +A[3] +A[5] +6*A[6];
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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