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