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

0,2

Comments

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

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

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A058487, A098151, A187100, A217771.

Sequence in context: A356728 A079089 A182705 * A261320 A179023 A321690

Adjacent sequences: A186921 A186922 A186923 * A186925 A186926 A186927

Keyword

nonn

Author

Michael Somos, Mar 05 2011

Status

approved