My father died last week and none of my family members contacted me to let me know. I found out a week later after the funeral services. I went online to find the obituary and I have also been left out of the surviving family members.
How do I find out for FREE if my father had a will?
It is obvious that I am trying to be forgotten as if I never existed.
Do I have any recourse?
I know that this is 2 questions, but I really need to know.
Thank you in advance for any help you can provide.

The only thing you can do is check to see if a will has been filed in your local probate court. But if one hasn't been filed, doesn't mean that there was no will. Depending on the nature of the estate they may be able to get out of filing.

If a will was entrusted to an attorney they will be required to inform you if you are mentioned. However, that may not have happened.

Unless you can find out who is the executor, there would be no one to sue.

Yes, you will have to ask other family members, call your fathers attorney, check with probate court.

One week is really early to worry. It could take months to start a probate case.

You posed the following question:
If he didn't leave the will, you would stand to inherit some share under intestacy laws in most states. In other words, your inheriting some share doesn't as much depend upon whether there was a will, but rather the terms of the will if there was one. So a better way to phrase the question is "how do I find out if there was a will disinheriting me?"

Do you know if your father left significant assets?

If so, and If they continue to ignore your questions, you could file a probate case on your own, asking to be appointed personal representative and alleging that, as far as you know, there was no will and that you want to administer the estate. That would give you the power to find out exactly what your father had when he died, and to assume control of those things

While AK's advice is sound, this would cause a battle between you and other relatives. In any case, given the circumstances, I would be prepared for a legal battle.

Are you estranged from all your relatives or just your immediate family?

Do you have any idea what assets your father had?

I'd start checking with the local Probate Court to see if the Will has been filed for probate. At this point, probably not because it's so soon.