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.
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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. |
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