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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.
0
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, 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, 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
OFFSET
1,2
COMMENTS
Column k lists the partial sums of the k-th column of triangle A237591.
We can see this sequence in the front view of the pyramid described in A245092.
EXAMPLE
Triangle begins:
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, 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;
99, 37, 20, 11, 4;
109, 41, 22, 13, 5;
120, 45, 24, 14, 7;
131, 49, 27, 15, 8, 1;
...
Illustration for n = 10:
We draw the first 10 rows of the infinite diagram described in A237591 as shown below:
Row _
1 _| |
2 _| _|
3 _| | |
4 _| _| |
5 _| | _|
6 _| _| | |
7 _| | | |
8 _| _| _| |
9 _| | | _|
10 |_ _ _ _ _ _|_ _|_|_|
Area 35 13 6 1
.
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].
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.
CROSSREFS
Row n has length A003056(n) hence column k starts in row A000217(k).
Row sums give A000217, n >= 1.
Column 1 gives A024206 without its initial zero.
Column 2 gives the partial sums of the A261348.
Sequence in context: A242390 A065395 A236631 * A367067 A340529 A197326
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Apr 13 2018
STATUS
approved