Pascal's triangle with 1s

Hello Math-Fun,

I had forgotten this, by Lars Blomberg (all terms are distinct):

> The triangle starts like this:

1

2 3

4 5 6

7 9 11 8

10 16 20 19 12

14 26 36 39 31 13

15 40 62 75 70 44 17

18 55 102 137 145 114 61 21

22 73 157 239 282 259 175 82 23

(...)

As I am preparing a paper for Tangente magazine about Pascal's triangle and the OEIS, I've had the idea of an artificial such triangle, based on the same adding rules, in which all terms must contain at least a digit 1.

I came to the solution below quite quickly (using Blomberg's "diamond" method) – but I'm sure this is not the lexicographiically earliest such triangle. Any taker?

... This "diamond shape" leads to this start:

A = 1, 10, 11, 81, 21, 91, 15, 102, 112, 701, 164, 117, 214, 813, 103, 1050, 281, 331, 1027, 916, 156,…

Best,
É.

__________

Maximilian H. was quick to correct the above triangle/diamond:

Nice idea!
But one should not forget that in Pascal's "triangle" (actually, an "infinite square matrix" or rather half-plane (column index in Z = (-oo .. +oo)), the first and last nonzero element of each row n >= 1 is not arbitrarily set  to 1 but as *all* elements, including those with indices k <= 0 and k >= n, they are given by the general rule   C(n+1, k) = C(n, k-1) + C(n, k).
Only row 0 is set to  C(0,k) = delta_{0,k}.

PS: With your rules, I get 13 and 121 as the first element of the 5th and
6th row (row index 4 and 5 if the first row has index 0). See the "Diamond" below

- Maximilian
                             1

                       10      11   
                   81      21      91   
               15     102     112     701   
           13     117     214     813     103   
      121     130     331     1027    916     151   
  817     251     461     1358    1943    1067     14   
     1068    712     1819    3301    3010    1081  
         1780    2531    5120    6311    4091  
             4311    7651   11431   10402  
                11962   19082   21833  
                    31044   40915  
                        71959  

S = [1, 10, 11, 81, 21, 91, 15, 102, 112, 701, 13, 117, 214, 813, 103, 121, 130, 331, 1027, 916, 151, 817, 251, 461, 1358, 1943, 1067, 14]