A186924 Expansion of (phi(-q^3) / phi(-q))^2 in powers of q where phi is a Ramanujan theta function.

1, 4, 12, 28, 60, 120, 228, 416, 732, 1252, 2088, 3408, 5460, 8600, 13344, 20424, 30876, 46152, 68268, 100016, 145224, 209120, 298800, 423840, 597108, 835804, 1162824, 1608508, 2212896, 3028632, 4124664, 5590976, 7544604, 10137264, 13565016

Offset: 0

Author

Michael Somos, Mar 05 2011

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 4*q + 12*q^2 + 28*q^3 + 60*q^4 + 120*q^5 + 228*q^6 + 416*q^7 + 732*q^8 + ...
		

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. A058487, A098151, A187100, A217771.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^2 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)
nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k))^2 / ((1-x^k)^2 * (1-x^(6*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)

PARI

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

Formula

Euler transform of period 6 sequence [ 4, 2, 0, 2, 4, 0, ...].

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A058487.

Convolution square of A098151. a(n) = 4 * A187100(n) unless n=0.

Convolution inverse of A217771. - Michael Somos, Sep 05 2015

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015