Let $m, n$ be any fixed natural numbers. Is it true that infinitely many elements of the sequence $\binom{m+k}{m}\_{k=1,2,3,...}$ ( as well as of the sequence $\left(\binom{m+k}{m}-1\right)\_{k=1,2,3,...})$ are representable as the sums of different elements of the sequence $\binom{n+k}{n}\_{k=1,2,3,...}$?



Representing one diagonal of Pascal's triangle using special sums coming from a different diagonal
For fixed $n$, the expression $\binom{n+k}n$ is a polynomial of degree $n$ in $k$ with no fixed prime divisor. A theorem of Kamke (referenced in the [first paragraph here](http://www.math.uiuc.edu/~ford/wwwpapers/warpoly.pdf)) says that there exists $x$ such that all sufficiently large integers are the sum of $s$ values of $\binom{n+k}n$. Of course this will include, for fixed $m$, all but finitely many values $\binom{m+j}m$.


