Do I need permission from the author for reusing data from a table?
We used a detailed data from a referenced paper as data to verify the formula which we derived. We included that paper as a reference, and changed the variable name of the table to our own. And we added the variable, which can be obtained from an easy calculation, to our data table. Also we used the original formula on the paper as a reference. Our manuscript has been already accepted and is soon to be published. Should we ask or inform the author of the referenced paper about our citation? If so, how should we do it?
If I understand correctly, you are worried about obtaining permission to re-publish an illustration (table) that has already been published by someone else. If you have used the data included in a table in a previous publication and are trying to re-publish the table with minimal changes, you will need to obtain permission from the journal publisher of the previous article. Journal websites have information on how to obtain permission for re-publishing illustrations. Once you have this permission, you will need to inform the journal which is about to publish your manuscript.
However, if you are not using the previously published table, and only using the data from the table for your own calculations to create your own table, then no permission is required. Only appropriate reference citation is required, which you have already done.
As for re-using a formula, it is sufficient to include an original formula from a previous paper as a reference, as long as you appropriately cite the reference in your manuscript.