A185146 Expansion of q^-2 * phi(q) * phi(q^4) / psi(q^8)^2 in powers of q where phi(), psi() are Ramanujan theta functions.

1, 2, 0, 0, 4, 4, 0, 0, 2, -2, 0, 0, -8, -4, 0, 0, -1, 6, 0, 0, 20, 4, 0, 0, -2, -8, 0, 0, -40, -8, 0, 0, 3, 10, 0, 0, 72, 16, 0, 0, 2, -16, 0, 0, -128, -20, 0, 0, -4, 22, 0, 0, 220, 24, 0, 0, -4, -30, 0, 0, -360, -36, 0, 0, 5, 44, 0, 0, 576, 52, 0, 0, 8

Offset

-2,2

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 = -2..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric W. Weisstein, MathWorld: Ramanujan Theta Functions

Formula

Expansion of eta(q^2)^5 * eta(q^8)^7 / (eta(q)^2 * eta(q^4)^4 * eta(q^16)^6) in powers of q.

Euler transform of period 16 sequence [ 2, -3, 2, 1, 2, -3, 2, -6, 2, -3, 2, 1, 2, -3, 2, 0, ...].

a(4*n) = a(4*n + 1) = 0. a(4*n - 2) = A029841(n). a(4*n - 1) = 2 * A029839(n).

Example

q^-2 + 2*q^-1 + 4*q^2 + 4*q^3 + 2*q^6 - 2*q^7 - 8*q^10 - 4*q^11 - q^14 + ...

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185146[n_] := SeriesCoefficient[(f[x, x]*f[x^4, x^4])/(x*f[x^8, x^24])^2, {x, 0, n}]; Table[A185146[n], {n, -2, 50}] (* G. C. Greubel, Jun 23 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<-2, 0, n += 2; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^4 * eta(x^16 + A)^6), n))}

Crossrefs

Cf. A029839, A029841.

Sequence in context: A341654 A072740 A226288 * A080964 A367054 A134014

Adjacent sequences: A185143 A185144 A185145 * A185147 A185148 A185149

Keyword

sign

Author

Michael Somos, Feb 28 2012

Status

approved