A128583 Expansion of chi(x) * psi(x^2) * phi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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

Offset: 0

Author

Michael Somos, Mar 11 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + x^7 + x^8 + 3*x^10 + ...
G.f. = q^5 + q^29 + q^53 + 2*q^77 + q^101 + 2*q^125 + q^149 + q^173 + ...
		

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. A128580, A128582, A190615, A229723, A259895, A260118.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] QPochhammer[ x^6]^2 / (QPochhammer[ x] QPochhammer[ x^12] ), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 4, 0, x^6] EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];

PARI

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

Formula

Expansion of q^(-5/24) * eta(q^2) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 6^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229723.

a(n) = A128582(4*n) = A259895(3*n) = A260118(4*n). 2 * a(n) = A190615(12*n + 2). - Michael Somos, Nov 15 2015

-2 * a(n) = A128580(12*n + 2). - Michael Somos, Dec 22 2016