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

0,4

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

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

Example

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 + ...

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}];

Prog

(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))};

Crossrefs

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

Sequence in context: A232624 A287368 A108149 * A218854 A172303 A064391

Adjacent sequences: A128580 A128581 A128582 * A128584 A128585 A128586

Keyword

nonn

Author

Michael Somos, Mar 11 2007

Status

approved