A279929 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.
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
Offset: 1
Examples
G.f. = q + q^2 + q^4 + 6*q^5 + 8*q^7 + q^8 + 6*q^10 + 12*q^11 + 14*q^13 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
Crossrefs
Cf. A281786.
Programs
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Magma
A := Basis( ModularForms( Gamma0(18), 2), 75); A[2] +A[3] +A[5] +6*A[6];
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Mathematica
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, 1, # == 3, 0, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)];
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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))))};
Formula
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.
From Amiram Eldar, Oct 23 2023: (Start)
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)
Comments