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A186924 Expansion of (phi(-q^3) / phi(-q))^2 in powers of q where phi is a Ramanujan theta function. 6
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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: A028346 A079089 A182705 * A261320 A179023 A321690

Adjacent sequences:  A186921 A186922 A186923 * A186925 A186926 A186927

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 05 2011

STATUS

approved

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Last modified October 17 14:18 EDT 2019. Contains 328113 sequences. (Running on oeis4.)