Let m > 0 be a number of terms of a polynomial and n ≥ 0 be a power the polynomial is raised to. The number of terms in the multinomial $(x_1 + x_2 + ... x_m)^n$ is given by ${n+m-1 \choose m-1}$ (number of multinomial coefficients). This makes up a single "floor" or "layer" in the pascal m-simplex.
For example, for $(x + y + z)^2$ we have m=3 and n=2, so the number of terms in its expansion is ${3+2 -1 \choose 3-1} = {5\choose2}= 10$ and indeed one can verify this with simple algebra.
My question is, what is the expression for the partial sum: $$\sum_{i=1}^{n}{i+m-1 \choose m-1}$$
This would correspond to the total number of terms in the expansion of: $$(x_1 + x_2 + ...x_m)^n + (x_1 + x_2 + ...x_m)^{n-1} + ..... + (x_1 + x_2 + ...x_m) + (x_1 + x_2 + ...x_m)^0$$
So basically I want to count the number of terms of all possible degrees up to n, in a multinomial with m different terms. This corresponds to counting the number of points in each "floor" of the pascal m-simplex up to a particular floor n, ignoring the values of the coefficients, just counting how many there are.
For $(x+y+z)^2$ we have: $1, x, y, z, xy, xz, yz, x^2, y^2, z^2$ so 10 terms, which can be seen in this image as the number of magenta, green, and black points (the first 3 layers) of the pascal 3-simplex
