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="/A263528/b263528.txt">Table of n, a(n) for n = 0..2500</a>

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allocated for Michael SomosExpansion of (psi(x) * psi(x^3) / f(-x^3)^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.

1, 2, 1, 8, 14, 6, 38, 60, 23, 140, 208, 76, 439, 626, 221, 1232, 1704, 584, 3182, 4300, 1443, 7700, 10212, 3368, 17673, 23076, 7497, 38808, 50008, 16046, 82070, 104560, 33190, 167996, 211920, 66628, 334202, 417902, 130288, 648224, 804254, 248858, 1229148

0,2

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

M. Somos, <a href="http://somos.crg4.com/multiq.html">Introduction to Ramanujan theta functions</a>

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

Expansion of q^(-1/2) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q) * eta(q^3)^3))^2 in powers of q.

Euler transform of period 6 sequence [ 2, -2, 8, -2, 2, 0, ...].

-2 * a(n) = A262930(2*n + 1).

G.f. = 1 + 2*x + x^2 + 8*x^3 + 14*x^4 + 6*x^5 + 38*x^6 + 60*x^7 + 23*x^8 + ...

G.f. = q + 2*q^3 + q^5 + 8*q^7 + 14*q^9 + 6*q^11 + 38*q^13 + 60*q^15 + 23*x^17 + ...

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)] / (4 QPochhammer[ x^3]^2))^2 / x, {x, 0, n}];

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

Cf. A262930.

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Michael Somos, Oct 19 2015

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