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A240061
Triangle read by rows, n>=1, 1<=k<=n. T(n,k) = number of cells in k-th row = number of cells in the k-th column of the diagram of the symmetric representation of sigma(n) in the first quadrant.
2
1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 0, 3, 1, 1, 1, 3, 2, 4, 1, 1, 1, 1, 0, 0, 4, 1, 1, 1, 1, 3, 2, 1, 5, 1, 1, 1, 1, 1, 1, 2, 0, 5, 1, 1, 1, 1, 1, 3, 1, 2, 1, 6, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 6, 1, 1, 1, 1, 1, 1, 4, 3, 4, 3, 1, 7, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 7
OFFSET
1,3
COMMENTS
Since the diagram is symmetric the number of cells in the k-th row equals the number of cells in k-th column, see example.
Row sums give A000203.
Right border gives A008619, n >= 1.
If n is an odd prime then row n lists (n+1)/2 ones, ((n+1)/2 - 2) zeros, and (n+1)/2.
EXAMPLE
Triangle begins:
1;
1, 2;
1, 1, 2;
1, 1, 2, 3;
1, 1, 1, 0, 3;
1, 1, 1, 3, 2, 4;
1, 1, 1, 1, 0, 0, 4;
1, 1, 1, 1, 3, 2, 1, 5;
1, 1, 1, 1, 1, 1, 2, 0, 5;
1, 1, 1, 1, 1, 3, 1, 2, 1, 6;
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 6;
1, 1, 1, 1, 1, 1, 4, 3, 4, 3, 1, 7;
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 7;
...
For n = 9 the symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
. Number of cells
._ _ _ _ _
|_ _ _ _ _| 5
. |_ _ 0
. |_ | 2
. |_|_ _ 1
. | | 1
. | | 1
. | | 1
. | | 1
. . . . . . . . |_| . . x 1
.
So the 9th row of triangle is [1, 1, 1, 1, 1, 1, 2, 0, 5].
For n = 9 and k = 7 there are two cells in the 7th row of the diagram, also there are two cells in the 7th column of the diagram, so T(9,7) = 2.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 26 2014
STATUS
approved