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%I #28 Jan 07 2025 10:12:31
%S 1,2,3,1,4,5,2,6,1,7,3,8,9,4,2,10,1,11,5,12,3,13,6,14,2,15,7,4,1,16,
%T 17,8,18,5,3,19,9,20,2,21,10,6,1,22,4,23,11,24,7,25,12,3,26,5,27,13,8,
%U 2,28,1,29,14,30,9,6,4,31,15,32,33,16,10,3,34,7,35,17,5,2,36,11,1,37,18,38,8,39,19,12,4,40,6,41,20
%N Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).
%C Row n gives the first A001227(n) terms of the n-th row of A379630.
%e Triangle begins:
%e 1;
%e 2;
%e 3, 1;
%e 4;
%e 5, 2;
%e 6, 1;
%e 7, 3;
%e 8;
%e 9, 4, 2;
%e 10, 1;
%e 11, 5;
%e 12, 3;
%e 13, 6;
%e 14, 2;
%e 15, 7, 4, 1;
%e 16;
%e 17, 8;
%e 18, 5, 3;
%e 19, 9;
%e 20, 2;
%e 21, 10, 6, 1;
%e 22, 4;
%e 23, 11;
%e 24, 7;
%e 25, 12, 3;
%e 26, 5;
%e 27, 13, 8, 2;
%e 28, 1;
%e ...
%e Illustration of initial terms:
%e _
%e _|1|
%e _|2 _|
%e _|3 |1|
%e _|4 _| |
%e _|5 |2 _|
%e _|6 _| |1|
%e _|7 |3 | |
%e _|8 _| _| |
%e _|9 |4 |2 _|
%e _|10 _| | |1|
%e _|11 |5 _| | |
%e _|12 _| |3 | |
%e _|13 |6 | _| |
%e _|14 _| _| |2 _|
%e _|15 |7 |4 | |1|
%e _|16 _| | | | |
%e _|17 |8 _| _| | |
%e _|18 _| |5 |3 | |
%e _|19 |9 | | _| |
%e _|20 _| _| | |2 _|
%e _|21 |10 |6 _| | |1|
%e _|22 _| | |4 | | |
%e _|23 |11 _| | | | |
%e _|24 _| |7 | _| | |
%e _|25 |12 | _| |3 | |
%e _|26 _| _| |5 | _| |
%e _|27 |13 |8 | | |2 _|
%e |28 | | | | | |1|
%e ...
%e The diagram is also the left part of the diagram of A379630.
%e The geometrical structure is the same as the diagram of A237591.
%Y Positive terms of A211343.
%Y Absolute values of A341971.
%Y Column 1 gives A000027.
%Y Right border gives A118235.
%Y Row lengths give A001227.
%Y Row sums give A286014.
%Y Subsequence of A286001 and of A299765 and of A379630.
%Y For the largest parts see A379633.
%Y Cf. A196020, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379631, A379634.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Dec 31 2024