Alternative way of writing the stars and bars formula where each bar is associated with at least one star.

I was looking for a different way of writing the formula of the number of different k-tuples of non-negative integers whose sum is equal to n and I thought of this formula followed by this combinatorial proof:

$\sum _{i=1}^{k}{\textstyle (}\genfrac{}{}{0ex}{}{n+1}{i}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{k-1}{i-1}{\textstyle )}$

The first combination is the number of positions that we can choose from to place the bars. There are $n+1$ positions to choose from, since now we can also place the bars before and after the stars. The next combination is the number of different ways of splitting the stars between the positions that were chosen for the bars to be placed.

Lastly, we'll have to add all the different ways of combinations of bar positions and number of bars in each position. I haven't found a flaw in my proof yet but I can't seem to conclude that the above formula is equal to

$(}\genfrac{}{}{0ex}{}{n+k-1}{k-1}{\textstyle )$

Could someone more experienced verify my proof or show me where the flaw is?

I was looking for a different way of writing the formula of the number of different k-tuples of non-negative integers whose sum is equal to n and I thought of this formula followed by this combinatorial proof:

$\sum _{i=1}^{k}{\textstyle (}\genfrac{}{}{0ex}{}{n+1}{i}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{k-1}{i-1}{\textstyle )}$

The first combination is the number of positions that we can choose from to place the bars. There are $n+1$ positions to choose from, since now we can also place the bars before and after the stars. The next combination is the number of different ways of splitting the stars between the positions that were chosen for the bars to be placed.

Lastly, we'll have to add all the different ways of combinations of bar positions and number of bars in each position. I haven't found a flaw in my proof yet but I can't seem to conclude that the above formula is equal to

$(}\genfrac{}{}{0ex}{}{n+k-1}{k-1}{\textstyle )$

Could someone more experienced verify my proof or show me where the flaw is?