A129438 Expansion of (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, 1, 0, 2, 2, 0, 0, 0, 2, 3, 0, 2, 4, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 6, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 6, 2, 0, 2, 4, 0, 0, 0, 0, 5, 0, 2, 0, 0, 0

Offset

0,4

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

Moebius transform is period 32 sequence [1, -1, 1, 2, -1, -1, -1, 0, 1, 1, 1, 2, -1, 1, -1, 0, 1, -1, 1, -2, -1, -1, -1, 0, 1, 1, 1, -2, -1, 1, -1, 0, ...].

a(4*n + 2) = a(8*n + 5) = a(8*n + 7) = 0.

a(n) = A125096(n) unless n=0. a(8*n + 1) = A112603(n). a(8*n + 3) = 2 * A033761(n).

a(2*n + 1) = A113411(n). a(4*n) = A033715(n). - Michael Somos, Nov 11 2015

Example

G.f. = 1 + q + 2*q^3 + 2*q^4 + 2*q^8 + 3*q^9 + 2*q^11 + 4*q^12 + 2*q^16 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] + EllipticTheta[ 4, 0, q^2] EllipticTheta[ 3, 0, q^4]) / 2, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])};

Crossrefs

Cf. A033715, A033761, A112603, A113411, A125096.

Sequence in context: A177338 A024158 A054978 * A125096 A037862 A337113

Adjacent sequences: A129435 A129436 A129437 * A129439 A129440 A129441

Keyword

nonn

Author

Michael Somos, Apr 14 2007

Status

approved