I'm Italian omg this sentence is horrible! We don't say "alcuno", but "nessuno", which means nothing, because we have this particular grammar structure called "doppia negazione" that implies that when you have to negate something, you have to do it twice. It's like saying "I haven't nothing",but in Italian is correct.
Check this out http://blogs.transparent.com/italian/tavolo-or-tavola/
According to this discussion http://www.duolingo.com/#/comment/224654 "tavolo" could be used but with a different preposition "sul tavolo".
Imagine a table, a piece of furniture. If you put something on this furniture is "sul tavolo", but if you create a decorated place, a space in which you eat, in which you place plates, decorations, as if you furnished a room, then this is no longer a piece of furniture, but your space to eat, just like a dining room . Then you use "IN TANOLA"
Total confusion after reading the comments. As I take the meaning: Someone has asked for a condiment, perhaps, and I reply "I do not have any on the table." Word by word, this comes out as 'do not' 'any' '(I) have' 'any' 'on' 'table'. Where did we find two negatives? Ne='any' or 'them'. Alcuno='any' or 'none'. So...is this literally 'do not' 'any' '(I) have' 'none' 'on' 'table'? This convoluted structure seems impossible to work out. How does the expression 'Non ne ho alcuno...' mean 'I do not have any...'? Is it simply colloquial Italian rather than a normal structure?