A028597 Expansion of theta_3(z) * theta_3(8*z) + theta_2(z) * theta_2(8*z) in powers of q^(1/4).

1, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0

Offset

0,5

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

Expansion of phi(q) * phi(q^32) - 2 * q * f(-q^8) * f(-q^16) = phi(q^4) * phi(q^32) + 4 * q^9 * phi(q^8) * psi(q^64) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jun 24 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 128^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 24 2011

Example

G.f. = 1 + 2*q^4 + 4*q^9 + 2*q^16 + 4*q^17 + 2*q^32 + 4*q^33 + 6*q^36 + 4*q^48 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^32] - 2 q QPochhammer[ q^8] QPochhammer[ q^16], {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=1, sqrtint( n), 2 * x^k^2, 1 + A) * sum( k=1, sqrtint( n\32), 2 * x^(32*k^2), 1 + A) - 2 * x * eta(x^8 + A) * eta(x^16 + A), n))}; /* Michael Somos, Jun 24 2011 */

Crossrefs

Sequence in context: A348271 A083804 A341840 * A353753 A028617 A261470

Adjacent sequences: A028594 A028595 A028596 * A028598 A028599 A028600

Keyword

nonn

Author

N. J. A. Sloane, Dec 11 1999

Status

approved