A259827 Expansion of phi(x) * f(-x^12)^3 / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, 0, 0, 3, 2, 0, 0, 4, 6, 0, 0, 4, 2, 0, 0, 4, 8, 0, 0, 7, 2, 0, 0, 8, 10, 0, 0, 4, 4, 0, 0, 5, 10, 0, 0, 8, 4, 0, 0, 12, 10, 0, 0, 8, 6, 0, 0, 4, 14, 0, 0, 12, 2, 0, 0, 8, 14, 0, 0, 8, 4, 0, 0, 9, 18, 0, 0, 12, 6, 0, 0, 16, 14, 0, 0, 4, 4, 0, 0, 12, 12

Offset

0,2

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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 phi(x) * c(x^4) / (3 * x^(4/3)) in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.

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

Euler transform of period 12 sequence [ 2, -3, 2, 0, 2, -3, 2, 0, 2, -3, 2, -3, ...].

a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A259655(n). 6 * a(n) = A259825(3*n + 4).

Example

G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 4*x^12 + 2*x^13 + ...
G.f. = q^4 + 2*q^7 + 3*q^16 + 2*q^19 + 4*q^28 + 6*q^31 + 4*q^40 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x^12]^3 / QPochhammer[ x^4], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};

Crossrefs

Cf. A259655, A259825.

Sequence in context: A261115 A216229 A224777 * A143161 A225853 A342128

Adjacent sequences: A259824 A259825 A259826 * A259828 A259829 A259830

Keyword

nonn

Author

Michael Somos, Jul 05 2015

Status

approved