A261321 Expansion of (phi(q) / phi(q^3))^2 in powers of q where phi() is a Ramanujan theta function.

1, 4, 4, -4, -12, -8, 12, 32, 20, -28, -72, -48, 60, 152, 96, -120, -300, -184, 228, 560, 344, -416, -1008, -608, 732, 1756, 1048, -1252, -2976, -1768, 2088, 4928, 2900, -3408, -7992, -4672, 5460, 12728, 7408, -8600, -19944, -11544, 13344, 30800, 17744, -20424

Offset: 0

Author

Michael Somos, Aug 14 2015

Keywords

• sign

Comments

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

The generating function is associated with a modular equation of degree 3 and is the multiplier denoted by "m". - Michael Somos, Nov 01 2017

Examples

			G.f. = 1 + 4*x + 4*x^2 - 4*x^3 - 12*x^4 - 8*x^5 + 12*x^6 + 32*x^7 + ...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 230 Entry 5(iii), g.f. denoted by multiplier m.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A123633, A128636, A139137, A187153, A213265, A217771, A228447, A261320.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3])^2, {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^3 + A)^4 * eta(x^12 + A)^4 / (eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^10), n))};

Formula

Expansion of eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4 / (eta(q)^4 * eta(q^4)^4 * eta(q^6)^10) in powers of q.

G.f.: (Sum_{k in Z} x^k^2) / (Sum_{k in Z} x^(3*k^2))^2.

a(n) = -(1)^n * A217771(n). a(n) = 4 * A187153(n) = 4 * A213265(n) unless n=0.

a(2*n) = 4 * A123633(n) = 4 * A128636(n) unless n=0. a(3*n) = -4 * A228447(n) unless n=0.

Convolution inverse is A261320. Convolution square of A139137.