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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.

%I A370386 #29 Mar 04 2024 08:51:29
%S A370386 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,
%T A370386 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,
%U A370386 12,8,9,10,12,14,9,10,11,13,16,11,12,13,15,20,8,9
%N 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.
%C A370386 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.
%e A370386 Triangle begins:
%e A370386   1;
%e A370386   2;
%e A370386   3, 4;
%e A370386   4, 5, 6, 5, 6, 8;
%e A370386   5, 6, 7, 8, 6, 7, 8, 10, 7, 8, 9, 12, 6, 7, 9, 10, 7, 8, 10, 12, 9, 10, 12, 16;
%e A370386   ...
%e A370386 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.
%e A370386 Tree begins:
%e A370386   row
%e A370386   [n]
%e A370386   [0]             1
%e A370386                   |
%e A370386   [1]         ____2____
%e A370386              /         \
%e A370386   [2]     __3__       __4__
%e A370386          /  |  \     /  |  \
%e A370386   [3]   4   5   6   5   6   8
%o A370386 (Python)
%o A370386 def A370386_rowlist(maxrow):
%o A370386     A,C = [[(1,)]],[[1]]
%o A370386     for i in range(maxrow):
%o A370386         A.append([])
%o A370386         C.append([])
%o A370386         for j in A[i]:
%o A370386             for k in j:
%o A370386                 x = j + (j[-1] + k,)
%o A370386                 A[i+1].append(x)
%o A370386                 C[i+1].append(x[-1])
%o A370386     return(C)
%Y A370386 The tree if only distinct values are allowed is A114622.
%Y A370386 Cf. A000142 (row lengths), A002720 (empirical row sums).
%Y A370386 Cf. A364673.
%K A370386 nonn,easy,tabf
%O A370386 0,2
%A A370386 _John Tyler Rascoe_, Feb 28 2024