A281731 Expansion of psi(-x) * psi(-x^3) / (phi(x^2) * phi(x^6)) in powers of x where phi(), psi() are Ramanujan theta functions.

1, -1, -2, 0, 5, 0, -10, 2, 18, 0, -32, 0, 55, -5, -90, 0, 144, 0, -226, 10, 346, 0, -522, 0, 777, -18, -1138, 0, 1648, 0, -2362, 32, 3348, 0, -4704, 0, 6554, -55, -9056, 0, 12425, 0, -16932, 90, 22922, 0, -30848, 0, 41282, -144, -54946, 0, 72768, 0, -95914

Offset

0,3

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

Formula

Euler transform of period 24 sequence [-1, -2, -2, 2, -1, -4, -1, 0, -2, -2, -1, 4, -1, -2, -2, 0, -1, -4, -1, 2, -2, -2, -1, 0, ...].

Expansion of (phi(q) / phi(q^4) - phi(q^3) / phi(q^12)) / (2*q) in powers of q^2 where phi() is a Ramanujan theta function.

Expansion of (psi(^2) / phi(^2))^2 - x * (psi(x^6) / phi(x^6))^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of chi(-x) * chi(-x^3) / (chi(x^2)^2 * chi(x^6)^2) = chi(-x) * psi(-x^3) / (chi(x^2)^2 * phi(x^6)) = psi(-x) * chi(-x^3) / (phi(x^2) * chi(x^6)^2) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of q^(-1/2) * eta(q) * eta(q^2) * eta(q^3) * eta(q^6) * eta(q^8)^2 * eta(q^24)^2 / (eta(q^4)^4 * eta(q^12)^4) in powers of q.

a(2*n) = - a(6*n + 1) = A079006(n). a(6*n + 3) = a(6*n + 5) = 0.

Example

G.f. = 1 - x - 2*x^2 + 5*x^4 - 10*x^6 + 2*x^7 + 18*x^8 - 32*x^10 + ...
G.f. = q - q^3 - 2*q^5 + 5*q^9 - 10*q^13 + 2*q^15 + 18*q^17 - 32*q^21 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/2) EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)]/(EllipticTheta[ 3, 0, x^2] EllipticTheta[ 3, 0, x^6]), {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) * eta(x^24 + A)^2 / (eta(x^4 + A)^4 * eta(x^12 + A)^4), n))};

Crossrefs

Cf. A079006.

Sequence in context: A242684 A242683 A242682 * A361921 A245928 A242839

Adjacent sequences: A281728 A281729 A281730 * A281732 A281733 A281734

Keyword

sign

Author

Michael Somos, Jan 28 2017

Status

approved