If I understand well, you consider the number $a(k,n,m)$ of multi-indices $a=(a_1,\dots,a_m)\in\{1,\dots,n\}^m$ with weight $\sum_{i=1}^m=k$. \sum_{i=1}^m a_i=k$. This is therefore the coefficient of$x^k$in $$\left (\sum _ {j=1}^n x^j \right)^m = x^m(1-x^n)^m (1-x)^{-m}\ .$$ Since the above generating function is the product of two functions with elementary power series expansion, a formula for$a(k,n,m)$is available as a convolution of binomial coefficients. Is this what you mean? This is certaily certainly in any text on the subject or so. 1 If I understand well, you consider the number$a(k,n,m)$of multi-indices$a=(a_1,\dots,a_m)\in\{1,\dots,n\}^m$with weight$\sum_{i=1}^m=k$. This is therefore the coefficient of$x^k$in $$\left (\sum _ {j=1}^n x^j \right)^m = x^m(1-x^n)^m (1-x)^{-m}\ .$$ Since the above generating function is the product of two functions with elementary power series expansion, a formula for$a(k,n,m)\$ is available as convolution of binomial coefficients. Is this what you mean? This is certaily in any text on the subject or so.