A246833 Expansion of psi(-x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

1, -2, 1, -2, 3, -2, 4, -4, 2, -2, 5, -4, 2, -6, 3, -6, 7, -2, 5, -4, 5, -6, 6, -2, 5, -10, 3, -6, 10, -4, 6, -8, 3, -8, 7, -6, 7, -6, 4, -6, 11, -6, 9, -10, 3, -6, 14, -4, 8, -10, 8, -10, 5, -6, 4, -16, 7, -4, 10, -4, 13, -14, 7, -8, 8, -6, 10, -12, 7, -12

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 q^(-3/4) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^2 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246815.

a(n) = (-1)^n * A213624(n). a(2*n) = A246832(n). a(2*n + 1) = -2 * A033763(n).

Example

G.f. = 1 - 2*x + x^2 - 2*x^3 + 3*x^4 - 2*x^5 + 4*x^6 - 4*x^7 + 2*x^8 - 2*x^9 + ...
G.f. = q^3 - 2*q^7 + q^11 - 2*q^15 + 3*q^19 - 2*q^23 + 4*q^27 - 4*q^31 + 2*q^35 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A033763, A213624, A246815, A246832.

Sequence in context: A364642 A029826 A192185 * A213624 A080845 A290370

Adjacent sequences: A246830 A246831 A246832 * A246834 A246835 A246836

Keyword

sign

Author

Michael Somos, Sep 04 2014

Status

approved