A260516 Expansion of f(x, x^2) * f(x^2, x^10) in powers of x where f(,) is Ramanujan's general theta function.

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

Offset

0,3

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-17/24) * eta(q^3)^2 * eta(q^4)^2 * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [ 1, 1, -1, -1, 1, -1, 1, 0, -1, 1, 1, -2, 1, 1, -1, 0, 1, -1, 1, -1, -1, 1, 1, -2, ...].

Example

G.f. = 1 + x + 2*x^2 + x^3 + x^4 + x^5 + 2*x^7 + x^9 + x^10 + x^11 + ...
G.f. = q^17 + q^41 + 2*q^65 + q^89 + q^113 + q^137 + 2*q^185 + q^233 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ -x^2, x^4] / (QPochhammer[ x, x^2] QPochhammer[ x^12, x^24]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3] / (x^(3/4) Sqrt[2] QPochhammer[ x]), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)^2*eta(q^24)/(eta(q)*eta(q^8)*eta(q^12))) \\ Altug Alkan, Aug 01 2018

Crossrefs

Sequence in context: A221650 A140883 A214021 * A064744 A135997 A026609

Adjacent sequences: A260513 A260514 A260515 * A260517 A260518 A260519

Keyword

nonn

Author

Michael Somos, Jul 27 2015

Status

approved