- Let $W(n, k, m)$ be an integer coefficients defined for $n > 0, 1 \leqslant k \leqslant n, m > 0$ with $W(n,k,m)=0$ for $n \leqslant 0$ or $k \leqslant 0$ such that
$$ W(n, k, m) = (k+m-1)W(n-1, k, m) + (n-k+1)W(n-1, k-1, m) + [m > 1]W(n, k, m-1), \\ W(1, 1, m) = \binom{m+1}{2} $$
For the related sequences in OEIS, see A008292, A062253, A062254, A062255. Note that offsets are the same only for A008292.
- Let
$$ T(n, k, m) = W(n-m+1, k-m+1, m) $$
I conjecture that for $n > 0, k > 0$
$$ \sum\limits_{i=1}^{n} T(n+k-1, n, i)(-1)^{n-i} = \binom{n+k-1}{k}. $$
Here is the PARI/GP program to check it numerically:
W(n, k, m) = if(n < 1 || k < 1, 0, if(n == 1 && k == 1, binomial(m+1, 2), (k+m-1)*W(n-1, k, m) + (n-k+1)*W(n-1, k-1, m) + if(m>1, W(n, k, m-1))))
T(n, k, m) = W(n-m+1, k-m+1, m)
test(n, k) = sum(i=1, n, T(n+k-1, n, i)*(-1)^(n-i)) == binomial(n+k-1, k)
Is there a way to prove it?
