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%I #22 Dec 25 2023 01:51:41
%S 1,6,12,6,-6,0,12,12,12,6,0,0,-6,12,24,0,-6,0,12,12,0,12,0,0,12,6,24,
%T 6,-12,0,0,12,12,0,0,0,-6,12,24,12,0,0,24,12,0,0,0,0,-6,18,12,0,-12,0,
%U 12,0,24,12,0,0,0,12,24,12,-6,0,0,12,0,0,0,0,12,12
%N Expansion of phi(x)^3 / phi(x^3) where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 227, Entry 4(iv).
%H G. C. Greubel, <a href="/A113660/b113660.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of a(q) + 2*q(q^2) - 2*a(q^4) = b(-q)^2 / b(q^2) = (b(q) - 2*b(q^4))^2 / b(q^2) = (c(q) + 2*c(q^4))^2 / (3 * c(q^2)) in powers of q where a(), b(), c() are cubic AGM functions. - _Michael Somos_, May 20 2015
%F Expansion of (eta(q^2)^15 * eta(q^3)^2 * eta(q^12)^2) / (eta(q)^6 * eta(q^4)^6 * eta(q^6)^5) in powers of q.
%F a(n) = 6*b(n) where b(n) is multiplicative with a(0) = 1, b(2^e) = (1 - 3(-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
%F Euler transform of period 12 sequence [ 6, -9, 4, -3, 6, -6, 6, -3, 4, -9, 6, -2, ...].
%F Moebius transform is period 12 sequence [ 6, 6, 0, -18, -6, 0, 6, 18, 0, -6, -6, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113973. - _Michael Somos_, May 20 2015
%F G.f.: 1 + 6 * ( Sum_{k>0} x^k / (1 + x^k + x^(2*k)) + 2*x^(4*k - 2) / (1 + x^(4*k - 2) + x^(8*k - 4)) ).
%F a(n) = 6 * A113661(n), if n>0.
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3) = 5.441398... (A304656). - _Amiram Eldar_, Dec 25 2023
%e G.f. = 1 + 6*q + 12*q^2 + 6*q^3 - 6*q^4 + 12*q^6 + 12*q^7 + 12*q^8 + 6*q^9 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* _Michael Somos_, May 20 2015 *)
%o (PARI) {a(n) = my(x); if( n<1, n==0, x = valuation(n, 2); if( n%2, 2, (1 - 3*(-1)^x))*3 * sumdiv(n/2^x, d, kronecker(-3, d)))};
%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 6*prod(k=1,matsize(A)[1], [p, e] = A[k,]; if( p==2, (1 - 3*(-1)^e) / 2, p==3, 1, p%6==1, e+1, !(e%2))))};
%o (PARI) {a(n) = if( n<1, n==0, 6 * direuler(p=2, n, if( p==2, 2 - (1 - 2*X) / (1 - X^2), 1 / ((1 - X) * (1 - kronecker(-3, p) * X))))[n])};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^15 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^6 * eta(x^6 + A)^5), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(12), 1), 74); A[1] + 6*A[2] + 12*A[3] + 6*A[4] - 6*A[5]; /* _Michael Somos_, May 20 2015 */
%Y Cf. A113661, A113973, A304656.
%Y Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
%K sign,easy
%O 0,2
%A _Michael Somos_, Nov 03 2005
%E Corrected by _Charles R Greathouse IV_, Sep 02 2009
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