[seqfan] Need help concerning a conjecture related to A122858 A229616 A282031

[Thomas Baruchel]

Hi,

with the help of my program, I could detect the following relation:

-3 A122858(n) - A229616(n) + 4 A282031(n) = 0

the exact log being:

> A122858 A229616 A282031    -->    -3 -1 4    (26)
> A122858 Expansion of E(k) * K(k) * (2/Pi)^2 in powers of q^2 where E(), K() are complete elliptic integrals and the nome q = exp( -Pi * K(k') / K(k)).
> A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.
> A282031 Coefficients in q-expansion of (9*E_2(q^3)-E_2(q))/8.

It may be interesting because the last sequence has no formula in its description.
However, I am not sure being able to get the full of this conjecture.

Some formulass like

   a(n)=if(n<1,n==0,sumdiv(n,d, -6*d* (-1)^d - 3* d*[0,1,-3,4,-3,1][d%6+1]));
   (PARI syntax)

are easy do build, but someone else could certainly get much more from the above
identity and probably add several lines in the description of A282031.

Best regards,

-- 
Thomas Baruchel