A370386 Irregular triangle read by rows. An infinite rooted tree having root node 1 in row n = 0. Nodes in row n each have n + 1 children with values m + k, where m is the value of the parent node and k takes the values of all nodes from the root to the parent including the parent itself.

1, 2, 3, 4, 4, 5, 6, 5, 6, 8, 5, 6, 7, 8, 6, 7, 8, 10, 7, 8, 9, 12, 6, 7, 9, 10, 7, 8, 10, 12, 9, 10, 12, 16, 6, 7, 8, 9, 10, 7, 8, 9, 10, 12, 8, 9, 10, 11, 14, 9, 10, 11, 12, 16, 7, 8, 9, 11, 12, 8, 9, 10, 12, 14, 9, 10, 11, 13, 16, 11, 12, 13, 15, 20, 8, 9

Offset: 0

Author

John Tyler Rascoe, Feb 28 2024

Keywords

• nonn
• easy
• tabf

Comments

The paths through the tree represent integer partitions which contain their own first differences and have least part 1. These partitions are counted, including those with any least part, in A364673.

Examples

			Triangle begins:
  1;
  2;
  3, 4;
  4, 5, 6, 5, 6, 8;
  5, 6, 7, 8, 6, 7, 8, 10, 7, 8, 9, 12, 6, 7, 9, 10, 7, 8, 10, 12, 9, 10, 12, 16;
  ...
The tree starts with root 1 in row n = 0. In row n = 2 the parent node 4 has 3 children using values of k: 1, 2, and 4.
Tree begins:
  row
  [n]
  [0]             1
                  |
  [1]         ____2____
             /         \
  [2]     __3__       __4__
         /  |  \     /  |  \
  [3]   4   5   6   5   6   8
		

Crossrefs

The tree if only distinct values are allowed is A114622.

Cf. A000142 (row lengths), A002720 (empirical row sums).

Cf. A364673.

Programs

Python

def A370386_rowlist(maxrow):
    A,C = [[(1,)]],[[1]]
    for i in range(maxrow):
        A.append([])
        C.append([])
        for j in A[i]:
            for k in j:
                x = j + (j[-1] + k,)
                A[i+1].append(x)
                C[i+1].append(x[-1])
    return(C)