I have come across a sequence and am wondering if anyone knows of a closed form expression for it. I am a bit too lazy to figure it out on my own seeing as it is such a minor part of what I am doing. Any help would be greatly appreciated.
The sequence is $c_{jk}$ such that $c_{(j+1)(k+1)} = c_{(j+1)k} - c_{jk}$ and $c_{j0} = 0$ for $j \neq 0$ and $c_{0k} = 1$ for $k\ge 0$
Making a small table of values gives the game away:
$$\begin{array}{c|rr} j\backslash k&0&1&2&3&4&5&6&7\\ \hline 0&1&1&1&1&1&1&1&1\\ 1&0&-1&-2&-3&-4&-5&-6&-7\\ 2&0&0&1&3&6&10&15&21\\ 3&0&0&0&-1&-4&-10&-20&-35\\ 4&0&0&0&0&1&5&15&35\\ 5&0&0&0&0&0&-1&-6&-21\\ 6&0&0&0&0&0&0&1&7\\ 7&0&0&0&0&0&0&0&-1 \end{array}$$
Clearly $$c_{jk}=(-1)^j\binom{k}j\;.$$
And indeed
$$(-1)^{j+1}\binom{k}{j+1}-(-1)^j\binom{k}j=(-1)^{j+1}\left(\binom{k}{j+1}+\binom{k}j\right)=(-1)^{j+1}\binom{j+1}{k+1}\;,$$
verifying the conjecture.