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Expansion of - c(-q) * c(q^2) / 9 in powers of q where c() is a cubic AGM theta function.
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%I #26 Jan 22 2024 00:15:25

%S 1,-1,3,-1,6,-3,8,-1,9,-6,12,-3,14,-8,18,-1,18,-9,20,-6,24,-12,24,-3,

%T 31,-14,27,-8,30,-18,32,-1,36,-18,48,-9,38,-20,42,-6,42,-24,44,-12,54,

%U -24,48,-3,57,-31,54,-14,54,-27,72,-8,60,-30,60,-18,62,-32,72

%N Expansion of - c(-q) * c(q^2) / 9 in powers of q where c() is a cubic AGM theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Number 91 of the 126 eta-quotients listed in Table 1 of Williams 2012.

%H G. C. Greubel, <a href="/A226132/b226132.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%H K. S. Williams, <a href="http://people.math.carleton.ca/~williams/papers/pdf/342.pdf">Fourier series of a class of eta quotients</a>, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

%F Expansion of (a(q) - a(q^2)) * (a(q^2) + 2 * a(q^4)) / 18 = c(q^2)^4 / (9 * c(q) * c(q^4)) = (b(-q) * b(q^2) - a(-q) * a(q^2)) / 9 in powers of q where a(), b(), c(q) are cubic AGM theta functions.

%F Expansion of q * (phi(q^3)^3 / phi(q)) * (ps(-q^3)^3 / psi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

%F Expansion of eta(q) * eta(q^4) * eta(q^6)^12 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^12)^3) in powers of q.

%F Euler transform of period 12 sequence [ -1, 3, 2, 2, -1, -6, -1, 2, 2, 3, -1, -4, ...].

%F Multiplicative with a(2^e) = -1 if e>0, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4/3 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226139.

%F G.f.: Sum_{k>0 not 3|k} x^k / (1 - (-x)^k)^2 = Sum_{k>0 not 2|k} k * x^k * (1 - x^k) / (1 + x^(3*k)).

%F G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 - x^(3*k))^6 * (1 + x^(3*k))^9 / ((1 - x^(2*k))^4 * (1 + x^(6*k))^3).

%F a(2*n) = - A121443(n). a(2*n + 1) = A185717(n).

%F a(n) = -(-1)^n * A121443(n). Convolution of A113447 and A113973.

%F From _Amiram Eldar_, Sep 12 2023: (Start)

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

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/54 = 0.18277045... . (End)

%e G.f. = q - q^2 + 3*q^3 - q^4 + 6*q^5 - 3*q^6 + 8*q^7 - q^8 + 9*q^9 - 6*q^10 + ...

%t a[ n_] := If[ n < 1, 0, Sum[ If[ OddQ[d] && ! Divisible[ n/d, 3], -d (-1)^(n/d), 0], {d, Divisors[ n]}]];

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

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, I q^(3/2)]^3 / EllipticTheta[ 2, 0, I q^(1/2)] / (4 (-1)^(1/4)), {q, 0, n}];

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, if(d%2 && (n/d)%3, -d * (-1)^(n/d))))};

%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -1, if( p==3, p^e, (p^(e+1) - 1) / (p - 1))))))};

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^12 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};

%Y Cf. A113447, A113973, A121443, A185717, A226139.

%Y Cf. A000122, A000700, A010054, A121373.

%K sign,easy,mult

%O 1,3

%A _Michael Somos_, May 27 2013