A grammatical explanation is that the noun phrase putting it on the table is the object of the verb remember.
Verbs can follow different patterns, or structures. Here are two ways for you to understand the structure of the verb remember:
- remember something/somebody
- remember doing something
You can see that I remember putting it on the table follows the second pattern, which is the most natural in this particular situation. However, there are other ways to use remember. For example, you can say:
- remember (that) something happened
Your sentence could possibly be: I remember (that) I put it on the table.
Like this, the verb remember does not use an object. Notice also that the verb put is past simple (I put), not present perfect (I've put) but that's a different point.
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