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="/A137828/b137828.txt">Table of n, a(n) for n = 0..1000</a>

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Expansion of phi(qx) / f(-qx^4)^2 in powers of q x where phi(), f() are Ramanujan theta functions.

1, 2, 0, 0, 4, 4, 0, 0, 9, 12, 0, 0, 20, 24, 0, 0, 42, 50, 0, 0, 80, 92, 0, 0, 147, 172, 0, 0, 260, 296, 0, 0, 445, 510, 0, 0, 744, 840, 0, 0, 1215, 1372, 0, 0, 1944, 2176, 0, 0, 3059, 3424, 0, 0, 4740, 5268, 0, 0, 7239, 8040, 0, 0, 10920, 12072, 0, 0, 16286, 17976, 0, 0

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 6^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 pi Pi i t) and g(t) is the g.f. for A051136.

a(4*n+2) = a(4*n+3) = 0.

a(4*n + 2) = a(4*n + 3) = 0.

a(4*n) = A051136(n). a(4*n + 1) = 2 * A137829(n).

G.f. = 1/q + 2*q^2 x + 4*qx^11 4 + 4*qx^14 5 + 9*qx^23 8 + 12*qx^26 9 + 20*qx^35 12 + 24*qx^13 + 42*x^38 16 + ...

G.f. = 1/q + 2*q^2 + 4*q^11 + 4*q^14 + 9*q^23 + 12*q^26 + 20*q^35 + 24*q^38 + ...

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)

(PARI) {a(n) = localmy(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x^4 + A)^4 / eta(x + A)^2, n))};

A051136(n) = a(4*n). 2 * A137829(n) = a(4*n+1).

Cf. A051136, A137829.

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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

_Michael Somos, _, Feb 12 2008