Question:summation of a simple series

Question:summation of a simple series

I would like to ask your assistance in understanding the following Maple interaction which speaks for itself.

```> sum(product(k+4-m, m=0..k-1)*z^k ,k=0..j);
/    j                 \
| --------'            |
|'  |  |               |  (j + 1)
|   |  |    (j + 5 - m)| z
|   |  |               |
|   |  |               |
\  m = 0               /               1
--------------------------------- - -------
4 z - 1                4 z - 1

> series(simplify(subs(j=6, %)), z=0, 16);
2       3        4         5         6            7            8
1 + 4 z + 16 z  + 64 z  + 256 z  + 1024 z  + 4096 z  - 1646816 z  - 6587264 z  -

9              10              11               12               13
26349056 z  - 105396224 z   - 421584896 z   - 1686339584 z   - 6745358336 z

14                 15      16
- 26981433344 z   - 107925733376 z   + O(z  )

> sum(product(k+4-m, m=0..k-1)*z^k ,k=0..6);
6          5         4        3       2
151200 z  + 15120 z  + 1680 z  + 210 z  + 30 z  + 5 z + 1

> sum((k+4)!/4!*z^k ,k=0..6);
6          5         4        3       2
151200 z  + 15120 z  + 1680 z  + 210 z  + 30 z  + 5 z + 1
```

This was with Maple 18 (X86 64 LINUX).

Best regards, Marko Riedel

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