Thread: n balls in d bins (non-standard?)

In how many ways can we distribute n balls over d bins where bin i contains no more balls than bin i-1 (i = 2,..., d)?

For instance with n = 4 balls and d = 3 bins, the solution is 4: (4, 0, 0), (3, 1, 0), (2, 2, 0) and (2, 1, 1). Matlab generated the table below (rows fix d, columns fix n).

Can anybody help me with an explicit formula?

Thanks in advance,
Sander

d\n	1	2	3	4	5	6	7	8	9	10
1	1	1	1	1	1	1	1	1	1	1
2	1	2	2	3	3	4	4	5	5	6
3	1	2	3	4	5	7	8	10	12	14
4	1	2	3	5	6	9	11	15	18	23
5	1	2	3	5	7	10	13	18	23	30
6	1	2	3	5	7	11	14	20	26	35
7	1	2	3	5	7	11	15	21	28	38
8	1	2	3	5	7	11	15	22	29	40
9	1	2	3	5	7	11	15	22	30	41
10	1	2	3	5	7	11	15	22	30	42

Frankly, I am confused by what you are asking.
This is a well know problem: the partitions of an integer into summands.
In order to program MathLab to produce that table, you must know that the function is recursively defined. So there is no simple closed form solution for this.