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A302800 Irregular triangle read by rows: T(n,k) is the area of the k-th region of the diagram with n rows described in A237591.

%I A302800 #34 Apr 16 2018 09:09:20
%S A302800 1,3,5,1,8,2,11,4,15,5,1,19,7,2,24,9,3,29,11,5,35,13,6,1,41,16,7,2,48,
%T A302800 18,9,3,55,21,11,4,63,24,12,6,71,27,14,7,1,80,30,16,8,2,89,34,18,9,3,
%U A302800 99,37,20,11,4,109,41,22,13,5,120,45,24,14,7,131,49,27,15,8,1,143,53,29,17,9,2
%N A302800 Irregular triangle read by rows: T(n,k) is the area of the k-th region of the diagram with n rows described in A237591.
%C A302800 Column k lists the partial sums of the k-th column of triangle A237591.
%C A302800 We can see this sequence in the front view of the pyramid described in A245092.
%e A302800 Triangle begins:
%e A302800     1;
%e A302800     3;
%e A302800     5,  1;
%e A302800     8,  2;
%e A302800    11,  4;
%e A302800    15,  5,  1;
%e A302800    19,  7,  2;
%e A302800    24,  9,  3;
%e A302800    29, 11,  5;
%e A302800    35, 13,  6,  1;
%e A302800    41, 16,  7,  2;
%e A302800    48, 18,  9,  3;
%e A302800    55, 21, 11,  4;
%e A302800    63, 24, 12,  6;
%e A302800    71, 27, 14,  7,  1;
%e A302800    80, 30, 16,  8,  2;
%e A302800    89, 34, 18,  9,  3;
%e A302800    99, 37, 20, 11,  4;
%e A302800   109, 41, 22, 13,  5;
%e A302800   120, 45, 24, 14,  7;
%e A302800   131, 49, 27, 15,  8,  1;
%e A302800 ...
%e A302800 Illustration for n = 10:
%e A302800 We draw the first 10 rows of the infinite diagram described in A237591 as shown below:
%e A302800 Row                           _
%e A302800 1                           _| |
%e A302800 2                         _|  _|
%e A302800 3                       _|   | |
%e A302800 4                     _|    _| |
%e A302800 5                   _|     |  _|
%e A302800 6                 _|      _| | |
%e A302800 7               _|       |   | |
%e A302800 8             _|        _|  _| |
%e A302800 9           _|         |   |  _|
%e A302800 10         |_ _ _ _ _ _|_ _|_|_|
%e A302800 Area             35     13  6 1
%e A302800 .
%e A302800 The diagram contains four regions and the areas of the successives regions from left to right are respectively [35, 13, 6, 1], so the 10th row of this triangle is [35, 13, 6, 1].
%e A302800 Note that this infinite diagram gives a correspondence between the number of partitions into k consecutive parts and the symmetric representation of A000203, A024916, A004125 and many other integer sequences. For more information see A196020, A236104, A235791, A237048, A237593, A262626, A286000 and A286001.
%Y A302800 Row n has length A003056(n) hence column k starts in row A000217(k).
%Y A302800 Row sums give A000217, n >= 1.
%Y A302800 Column 1 gives A024206 without its initial zero.
%Y A302800 Column 2 gives the partial sums of the A261348.
%Y A302800 Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A286000, A286001.
%K A302800 nonn,tabf
%O A302800 1,2
%A A302800 _Omar E. Pol_, Apr 13 2018