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

Author

Michael Somos, May 31 2013

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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

Crossrefs

Cf. A033715, A112603, A113411, A242609.

Programs

Mathematica

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

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))};

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