A261115 Expansion of f(x, x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

1, 2, 0, 0, 3, 2, 0, 0, 3, 4, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 4, 2, 0, 0, 1, 6, 0, 0, 2, 2, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 1, 4, 0, 0, 2, 4, 0, 0, 2, 4, 0, 0, 1, 2, 0, 0, 8, 0, 0, 0, 2, 4, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 4, 4, 0

Offset

0,2

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^(-1/6) * eta(q^2)^5 * eta(q^8) * eta(q^12)^2 / (eta(q)^2 * eta(q^4)^3 * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -2, ...].

a(n) = (-1)^n * A260110(n) = A128580(3*n) = A129402(3*n) = A115660(6*n + 1) = A128581(6*n + 1) = A192013(6*n + 1).

a(4*n) = A113780(n). a(4*n + 1) = 2 * A260089(n). a(4*n + 2) = a(4*n + 3) = 0.

Example

G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 3*x^8 + 4*x^9 + 2*x^12 + 2*x^13 + 2*x^16 + ...
G.f. = q + 2*q^7 + 3*q^25 + 2*q^31 + 3*q^49 + 4*q^55 + 2*q^73 + 2*q^79 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^12] / 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)^5 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^24 + A)), n))};

Crossrefs

Cf. A113780, A115660, A128580, A128581, A129402, A192013, A260110.

Sequence in context: A125095 A113411 A260110 * A216229 A224777 A259827

Adjacent sequences: A261112 A261113 A261114 * A261116 A261117 A261118

Keyword

nonn

Author

Michael Somos, Aug 08 2015

Status

approved