[seqfan] A conjecture about the numerator of Euler(2n-1,n)

[Vladimir Shevelev]

Dear SeqFans,

Now I have just submitted a new sequence
A291897 "Numerator of Euler(2n-1,n)". 
I proved that 
a(n)=2(-1)^n*A006519(2n)*(1^(2n-1) - 2^(2n-1)+
...+(-1)^n*(n-1)^(2n-1)) + A002425(n).
The sequence begins 1, 9, 125, 32977,...
My observation is about an interesting property:
a(n) is divisible by (2*n-1)^2 (conjecture).
Note that sometimes, maybe, for prime n of the form
4*k+1, a(n) is divisible by (2n-1)^3 ( for example, for
 n=1,5,13,17,...).
Can anyone analyze this property?

Best regards,
Vladimir