A230278 Expansion of q^(-2/3) * eta(q^2)^10 / eta(q)^4 in powers of q.

1, 4, 4, 0, 0, -8, -16, 0, -10, -20, 16, 0, 0, 40, 0, 0, 39, 28, 0, 0, 0, -40, 32, 0, -70, 0, -64, 0, 0, -80, 0, 0, 49, -20, -40, 0, 0, 112, 80, 0, -22, 56, 64, 0, 0, 88, 0, 0, 110, -140, 0, 0, 0, 0, -160, 0, -128, 52, 0, 0, 0, -280, 0, 0, -130, 28, 156, 0, 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 psi(x^2)^2 * f(x)^4 = phi(x)^2 * f(-x^4)^4 = psi(x)^4 * f(-x^2)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Euler transform of period 2 sequence [ 4, -6, ...].

G.f.: Product_{k>0} (1 - x^k)^6 * (1 + x^k)^10.

a(4*n + 3) = a(8*n + 4) = 0. 2 * a(n) = A230277(3*n + 2).

Convolution square of A113277.

Example

G.f. = 1 + 4*x + 4*x^2 - 8*x^5 - 16*x^6 - 10*x^8 - 20*x^9 + 16*x^10 + ...
G.f. = q^2 + 4*q^5 + 4*q^8 - 8*q^17 - 16*q^20 - 10*q^26 - 20*q^29 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^10 / QPochhammer[ q]^4, {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 / eta(x + A)^4, n))}

Crossrefs

Cf. A113277, A230277.

Sequence in context: A291696 A291649 A216060 * A190113 A165727 A284609

Adjacent sequences: A230275 A230276 A230277 * A230279 A230280 A230281

Keyword

sign

Author

Michael Somos, Oct 15 2013

Status

approved