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A330379
Triangle read by rows: T(n,k) (1 <= k <= n) is the sum of the sizes of all right angles of size k of all partitions of n.
3
1, 0, 4, 0, 0, 9, 1, 0, 3, 16, 2, 0, 0, 8, 25, 3, 4, 0, 8, 15, 36, 4, 8, 0, 0, 20, 24, 49, 5, 12, 9, 0, 15, 36, 35, 64, 7, 16, 21, 0, 5, 36, 56, 48, 81, 9, 20, 33, 16, 0, 36, 63, 80, 63, 100, 13, 24, 45, 40, 0, 12, 77, 96, 108, 80, 121
OFFSET
1,3
COMMENTS
Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).
REFERENCES
G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143.
FORMULA
T(n,k) = k*A330369(n,k).
EXAMPLE
Triangle begins:
1;
0, 4;
0, 0, 9;
1, 0, 3, 16;
2, 0, 0, 8, 25;
3, 4, 0, 8, 15, 36;
4, 8, 0, 0, 20, 24, 49;
5, 12, 9, 0, 15, 36, 35, 64;
7, 16, 21, 0, 5, 36, 56, 48, 81;
9, 20, 33, 16, 0, 36, 63, 80, 63, 100;
13, 24, 45, 40, 0, 12, 77, 96, 108, 80, 121;
...
Below the figure 1 shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. The figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.
.
. Right-angles Right
Part Ferrers diagram Part diagram angle
_ _ _ _ _ _ _
7 * * * * * * * 7 | _ _ _ _ _ _| 14
6 * * * * * * 6 | | _ _ _ _| 8
3 * * * 3 | | | | 2
3 * * * 3 | | |_|
2 * * 2 | |_|
1 * 1 | |
1 * 1 | |
1 * 1 |_|
.
Figure 1. Figure 2.
.
For n = 8 the partitions of 8 and their respective right-angles diagrams look as shown below:
.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1| |8 2| _|8 3| _ _|8 4| _ _ _|8 5| _ _ _ _|8
1| | 1| | 1| | 1| | 1| |
1| | 1| | 1| | 1| | 1| |
1| | 1| | 1| | 1| | 1|_|
1| | 1| | 1| | 1|_|
1| | 1| | 1|_|
1| | 1|_|
1|_|
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
6| _ _ _ _ _|8 7| _ _ _ _ _ _|8 8|_ _ _ _ _ _ _ _|8
1| | 1|_|
1|_|
.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
2| _|7 3| _ _|7 4| _ _ _|7 5| _ _ _ _|7 6| _ _ _ _ _|7
2| |_|1 2| |_| 1 2| |_| 1 2| |_| 1 2|_|_| 1
1| | 1| | 1| | 1|_|
1| | 1| | 1|_|
1| | 1|_|
1|_|
.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
2| _|6 3| _ _|6 3| _ _|6 4| _ _ _|6 4| _ _ _|6 5| _ _ _ _|6
2| | |2 2| | | 2 3| |_ _|2 2| | | 2 3| |_ _| 2 3|_|_ _| 2
2| |_| 2| |_| 1| | 2|_|_| 1|_|
1| | 1|_| 1|_|
1|_|
.
_ _ _ _ _ _ _ _ _
2| _|5 3| _ _|5 4| _ _ _|5
2| | |3 3| | _|3 4|_|_ _ _|3
2| | | 2|_|_|
2|_|_|
.
There are 5 right angles of size 1, so T(8,1) = 5*1 = 5.
There are 6 right angles of size 2, so T(8,2) = 6*2 = 12.
There are 3 right angles of size 3, so T(8,3) = 3*3 = 9.
There are no right angle of size 4, so T(8,4) = 0*4 = 0.
There are 3 right angles of size 5, so T(8,5) = 3*5 = 15.
There are 6 right angles of size 6, so T(8,6) = 6*6 = 36.
There are 5 right angles of size 7, so T(8,7) = 5*7 = 35.
There are 8 right angles of size 8, so T(8,8) = 8*8 = 64.
Hence the 8th row of triangle is [5, 12, 9, 0, 15, 36, 35, 64].
The row sum gives A066186(8) = 8*A000041(8) = 8*22 = 176.
CROSSREFS
Row sums give A066186, n >= 1.
Row sums of the terms that are after last zero give A179862.
Cf. A188674.
Sequence in context: A048728 A002915 A127733 * A364102 A249035 A244121
KEYWORD
nonn,tabl,more
AUTHOR
Omar E. Pol, Dec 31 2019
STATUS
approved