A257398 Expansion of phi(-x^6)^2 / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

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

Expansion of phi(x^3) * f(x, x^2) in powers of x where phi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

Expansion of q^(-1/24) * eta(q^2) * eta(q^6)^4 / (eta(q) * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [1, 0, 1, 0, 1, -4, 1, 0, 1, 0, 1, -2, ...].

Example

G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + x^7 + 2*x^8 + 2*x^10 + ...
G.f. = q + q^25 + q^49 + 2*q^73 + 2*q^97 + 3*q^121 + q^169 + 2*q^193 + ...

Mathematica

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

Crossrefs

Sequence in context: A160115 A139365 A071479 * A182631 A231728 A303545

Adjacent sequences: A257395 A257396 A257397 * A257399 A257400 A257401

Keyword

nonn,changed

Author

Michael Somos, Apr 21 2015

Status

approved