A260415 Expansion of f(x, x^2) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

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

Offset

0,6

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^(-5/24) * eta(q^2) * eta(q^3)^2 * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^24)) in powers of q.

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

Example

G.f. = 1 + x + x^2 + x^4 + 2*x^5 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + x^11 + ...
G.f. = q^5 + q^29 + q^53 + q^101 + 2*q^125 + q^149 + q^173 + q^197 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x] QPochhammer[ x^3]^2 QPochhammer[ -x^6, x^12] / (2^(1/2) x^(1/4) QPochhammer[ x]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^12] / (QPochhammer[ x, x^2] QPochhammer[ x^4, x^8]), {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^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)), n))};

Crossrefs

Sequence in context: A221362 A114116 A054532 * A370942 A214710 A120888

Adjacent sequences: A260412 A260413 A260414 * A260416 A260417 A260418

Keyword

sign

Author

Michael Somos, Jul 27 2015

Status

approved