Iterated Difference Sequences

Consider the following list of sequences:

• Sequence $1$: $\quad 1, 1, 1, 1, 1, 1, 1, 1, \dots$
• Sequence $2$: $\quad 1, 2, 3, 4, 5, 6, 7, 8, \dots$
• Sequence $3$: $\quad 1, 2, 4, 7, 11, 16, 22, 29, \dots$
• Sequence $4$: $\quad 1, 2, 4, 8, 15, 26, 42, 64, \dots$
• $\cdots$

In general, the differences between consecutive terms of sequence $m$ form exactly sequence $m-1$. Find the general term of sequence $m$, and the sum of its first $n$ terms.