A226225 Expansion of phi(q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 2, 8, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 6, 4, 0, 0, 4, 0, 0, 0, 0, 10, 0, 0, 0, 0

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q^2) * eta(q^16))^5 / (eta(q) * eta(q^4) * eta(q^8) * eta(q^32))^2 in powers of q.

Euler transform of period 32 sequence [2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -4, 2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} (x^8)^k^2).

a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0. a(4*n) = a(8*n) = A033715(n). a(4*n + 1) = A033715(4*n + 1). a(8*n + 1) = 2 * A112603(n). a(8*n + 4) = 2 * A113411(n).

(-1)^n * a(n) = A242609(n). - Michael Somos, Feb 20 2015

Example

G.f. = 1 + 2*q + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^12 + 2*q^16 + 4*q^17 + 4*q^24 + 2*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, n==0, 2 * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^16 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^8 + A) * eta(x^32 + A))^2, n))};

Crossrefs

Cf. A033715, A112603, A113411, A242609.

Sequence in context: A108498 A178923 A242609 * A329265 A130209 A109127

Adjacent sequences: A226222 A226223 A226224 * A226226 A226227 A226228

Keyword

nonn

Author

Michael Somos, May 31 2013

Status

approved