Hello, I've gotten far on this exercise, with the following insight:
Here is a matrix of examples (vertical-axis is n=1,2,3,4,5,6,7,8; horizontal-axis is k=1,2,3,4)
1: 1 1 1 1
2: 2 3 4 5
3: 3 6 10 15
4: 4 10 20 35
5: 5 15 35 70
6: 6 21 56 126
7: 7 28 84 210
8: 8 36 120 330
Now, there is an obvious pattern among the numbers, being triangular numbers, sum of triangular numbers, sum of sum of triangular numbers, and so on. My question is:
Can you help me find closed form expressions for the sums of the entries:
- Down the columns
- Across the rows
- And for general n,k?
Thanks!