(48G/50g) Binomial Transform, Difference Table

[Thomas Klemm]

That's just this formula for the inverse binomial transform from the linked Wikipedia article:

\(a_n=\sum_{k=0}^n {n\choose k} t_k\)

In the given example it leads to:

\(\begin{align*}
a_n&=0\binom{n}{0}+1\binom{n}{1}+2\binom{n}{2}+1\binom{n}{3}\\
&= 0+1\frac{n}{1}+2\frac{n(n-1)}{2}+1\frac{n(n-1)(n-2)}{6}\\
&=n+ n(n-1)+\frac{n(n-1)(n-2)}{6}\\
&=\frac{n(n+1)(n+2)}{6}
\end{align*}\)

This is in accordance with A000292.

Cheers
Thomas