Power of a Jordan block

A Jordan block of size $k$ with eigenvalue $λ$ :

J_{k} (λ) = λ I + N, N^{k} = 0

General formula:

J_{k} (λ)^{m} = \sum_{r = 0}^{k - 1} (\binom{m}{r}) λ^{m - r} N^{r}

Entrywise:

(J_{k} (λ)^{m})_{i j} = {\begin{cases} (\binom{m}{j - i}) λ^{m - (j - i)} & if i \leq j, \\ 0 & if i > j \end{cases}

Particular cases:

J_{2} (λ)^{m} = (\begin{matrix} λ^{m} & m λ^{m - 1} \\ 0 & λ^{m} \end{matrix})

J_{3} (λ)^{m} = (\begin{matrix} λ^{m} & m λ^{m - 1} & (\binom{m}{2}) λ^{m - 2} \\ 0 & λ^{m} & m λ^{m - 1} \\ 0 & 0 & λ^{m} \end{matrix})

J_{4} (λ)^{m} = (\begin{matrix} λ^{m} & m λ^{m - 1} & (\binom{m}{2}) λ^{m - 2} & (\binom{m}{3}) λ^{m - 3} \\ 0 & λ^{m} & m λ^{m - 1} & (\binom{m}{2}) λ^{m - 2} \\ 0 & 0 & λ^{m} & m λ^{m - 1} \\ 0 & 0 & 0 & λ^{m} \end{matrix})