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

M. Somos, <a href="http://somos.crg4.comA010815/multiqa010815.htmltxt">Introduction to Ramanujan theta functions</a>

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G. C. Greubel, <a href="/A132973/b132973.txt">Table of n, a(n) for n = 0..1000</a>

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1, -3, 3, -3, 3, 0, 3, -6, 3, -3, 0, 0, 3, -6, 6, 0, 3, 0, 3, -6, 0, -6, 0, 0, 3, -3, 6, -3, 6, 0, 0, -6, 3, 0, 0, 0, 3, -6, 6, -6, 0, 0, 6, -6, 0, 0, 0, 0, 3, -9, 3, 0, 6, 0, 3, 0, 6, -6, 0, 0, 0, -6, 6, -6, 3, 0, 0, -6, 0, 0, 0, 0, 3, -6, 6, -3, 6, 0, 6, -6, 0, -3, 0, 0, 6, 0, 6, 0, 0, 0, 0, -12, 0, -6, 0, 0, 3, -6, 9, 0, 3, 0, 0, -6

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

M. Somos, <a href="http://cis.csuohio.edu/~somos.crg4.com/multiq.pdfhtml">Introduction to Ramanujan theta functions</a>

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

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 Pi i t) and g(t) is the g.f. for A113447.

a(6*n+5) = 0.

a(n) = (-1)^n * a(n) = A107760(n). Convolution inverse of A132974.

a(2*n) = A107760(n). a(2*n + 1) = -3 * A033762(n). a(3*n) = A132973(n). a(3*n + 1) = -3 * A227696(n). - Michael Somos, Oct 31 2015

a(6*n + 1) = -3 * A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. - Michael Somos, Oct 31 2015

G.f. = 1 - 3*q + 3*q^2 - 3*q^3 + 3*q^4 + 3*q^6 - 6*q^7 + 3*q^8 - 3*q^9 + 3*q^12 + ...

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]/2, {q, 0, n}] ; (* Michael Somos, May 26 2013 *)

(PARI) {a(n) = if( n<1, n==0, 3 * (-1)^n * sumdiv(n, d, kronecker(-12, d)))};

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

Cf. A033687, A097195, A107760, A132973, A132974, A227696.

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_Michael Somos, _, Sep 07 2007

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