A123884 Expansion of phi(x) * phi(-x^6) / chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

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

Offset: 0

Author

Michael Somos, Oct 17 2006

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 2*q^73 + 2*q^97 + 2*q^109 + ...
		

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

Crossrefs

Cf. A093829, A131961, A131962, A248886.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))};

Formula

Expansion of q^(-1/12) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^4) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -2, ...].

a(n) = A093829(12*n + 1).

a(n) = (-1)^n * A248886(n). a(2*n) = A131961(n). a(2*n + 1) = 2 * A131963(n). - Michael Somos, Oct 01 2015