# How to control counters generating a matrix with lambda?

I am trying to generate this famous matrix without the expected diagonal of 1's. So the expected output is as follows:

```
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
[1,] 1 0 0 0 0 0 0 0 0 0 0
[2,] 1 2 0 0 0 0 0 0 0 0 0
[3,] 1 3 3 0 0 0 0 0 0 0 0
[4,] 1 4 6 4 0 0 0 0 0 0 0
[5,] 1 5 10 10 5 0 0 0 0 0 0
[6,] 1 6 15 20 15 6 0 0 0 0 0
[7,] 1 7 21 35 35 21 7 0 0 0 0
[8,] 1 8 28 56 70 56 28 8 0 0 0
[9,] 1 9 36 84 126 126 84 36 9 0 0
[10,] 1 10 45 120 210 252 210 120 45 10 0
[11,] 1 11 55 165 330 462 462 330 165 55 11
```

I generated it with the following loop in R:

```
n = 11
asymmetric.lower.triang.Pascal = matrix(0,n,n)
for(i in 1:n){
for(k in 0:(i-1))
asymmetric.lower.triang.Pascal[i,k+1] = choose(i,k)
}
```

I can generate the Pascal matrix in SageMath with the loop:

```
r = 10
matrix(ZZ, r, lambda i, j: binomial(i, j))
[ 1 0 0 0 0 0 0 0 0 0]
[ 1 1 0 0 0 0 0 0 0 0]
[ 1 2 1 0 0 0 0 0 0 0]
[ 1 3 3 1 0 0 0 0 0 0]
[ 1 4 6 4 1 0 0 0 0 0]
[ 1 5 10 10 5 1 0 0 0 0]
[ 1 6 15 20 15 6 1 0 0 0]
[ 1 7 21 35 35 21 7 1 0 0]
[ 1 8 28 56 70 56 28 8 1 0]
[ 1 9 36 84 126 126 84 36 9 1]
```

but I want to get rid of the diagonal of 1's with the same manipulation of the counters or indices I used above. I have no idea what `lambda`

does, having found the basic code for matrix construction in SageMath online. So the question amounts to whether I could make `i`

run from `1`

to `n`

while `j`

runs from `0`

to `n - 1`

or viceversa.

"I have no idea what lambda does" - it means you need to learn some Ptthon. E.g. "Think Python" may be a good starting point: https://greenteapress.com/thinkpython...