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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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%I #17 Oct 23 2023 02:00:54

%S 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,

%T 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,

%U 30,60,0,62,32,0,1,84,0,68,18,0,48,72,0,74,38,0

%N 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.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A279929/b279929.txt">Table of n, a(n) for n = 1..5000</a>

%F 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.

%F 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.

%F 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.

%F 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.

%F a(2*n) = a(n). a(3*n) = 0.

%F From _Amiram Eldar_, Oct 23 2023: (Start)

%F Dirichlet g.f.: zeta(s-1) * zeta(s) * (1 - 2^(1-s)) * (1 - 3^(1-s)) * (1 - 3^(-s)).

%F Sum_{k=1..n} a(k) ~ (2*Pi^2/81) * n^2. (End)

%e G.f. = q + q^2 + q^4 + 6*q^5 + 8*q^7 + q^8 + 6*q^10 + 12*q^11 + 14*q^13 + ...

%t a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, 1, # == 3, 0, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)];

%o (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))};

%o (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))))};

%o (Magma) A := Basis( ModularForms( Gamma0(18), 2), 75); A[2] +A[3] +A[5] +6*A[6];

%Y Cf. A281786.

%K nonn,mult

%O 1,5

%A _Michael Somos_, Apr 11 2017