A260186 Expansion of (phi(q^4) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

1, -4, 12, -32, 80, -184, 400, -832, 1664, -3220, 6056, -11104, 19904, -34968, 60320, -102336, 171008, -281800, 458428, -736928, 1171552, -1843328, 2872368, -4435392, 6790656, -10313180, 15544136, -23259968, 34568576, -51042392, 74901984, -109268224, 158507008

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q) / eta(q^16))^4 * (eta(q^8) / eta(q^2))^10 in powers of q.

Euler transform of period 16 sequence [ -4, 6, -4, 6, -4, 6, -4, -4, -4, 6, -4, 6, -4, 6, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A216060.

Convolution inverse is A216060. Convolution square of A112128.

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

G.f. = 1 - 4*x + 12*x^2 - 32*x^3 + 80*x^4 - 184*x^5 + 400*x^6 - 832*x^7 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^16 + A))^4 * (eta(x^8 + A) / eta(x^2 + A))^10, n))};

Crossrefs

Cf. A112128, A216060.

Sequence in context: A097392 A373046 A090634 * A085750 A001787 A118442

Adjacent sequences: A260183 A260184 A260185 * A260187 A260188 A260189

Keyword

sign

Author

Michael Somos, Jul 17 2015

Status

approved