A230442 Expansion of q^(-1/6) * eta(q)^2 * eta(q^2) in powers of q.

1, -2, -2, 4, 1, 2, -2, -4, -1, -4, 6, 0, 0, 6, 4, -4, -4, 2, -6, 0, -5, 2, 0, 0, 4, 2, 6, 4, -1, -6, 2, 0, 4, -6, -8, -8, 8, -2, -6, 8, -4, 4, 4, 4, 4, -2, -2, 8, -1, 4, -4, 0, -4, -8, -6, 0, 0, 0, 6, -8, -3, -2, 6, -4, 8, 12, -2, -4, 4, 0, 10, 4, -4, -2, 0, -8, -4, -2, 4, 4, -12, 2, -4, 0, -12, 4, -4

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 f(-x)^3 / chi(-x) = f(-x^2)^3 * chi(-x)^2 = phi(-x) * f(-x^2)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

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

G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k)).

Convolution square is A230280.

Example

G.f. = 1 - 2*x - 2*x^2 + 4*x^3 + x^4 + 2*x^5 - 2*x^6 - 4*x^7 - x^8 + ...
G.f. = q - 2*q^7 - 2*q^13 + 4*q^19 + q^25 + 2*q^31 - 2*q^37 - 4*q^43 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^2], {x, 0, n}];

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^2)) \\ Altug Alkan, Apr 18 2018

Crossrefs

Cf. A230280.

Sequence in context: A245204 A099320 A206714 * A034951 A317826 A317836

Adjacent sequences: A230439 A230440 A230441 * A230443 A230444 A230445

Keyword

sign

Author

Michael Somos, Oct 18 2013

Extensions

Terms and offset corrected by Joerg Arndt, Oct 21 2013

Status

approved