A210952
Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the partitions of n but with the partitions aligned to the right margin.
Original entry on oeis.org
1, 1, 3, 1, 3, 5, 1, 3, 7, 9, 1, 3, 7, 12, 12, 1, 3, 7, 14, 21, 20, 1, 3, 7, 14, 24, 31, 25, 1, 3, 7, 14, 26, 40, 47, 38, 1, 3, 7, 14, 26, 43, 61, 66, 49, 1, 3, 7, 14, 26, 45, 70, 92, 93, 69, 1, 3, 7, 14, 26, 45, 73, 106, 130, 124, 87, 1, 3, 7, 14
Offset: 1
For n = 6 the illustration shows the partitions of 6 aligned to the right margin and below the sums of the columns:
.
. 6
. 3 + 3
. 4 + 2
. 2 + 2 + 2
. 5 + 1
. 3 + 2 + 1
. 4 + 1 + 1
. 2 + 2 + 1 + 1
. 3 + 1 + 1 + 1
. 2 + 1 + 1 + 1 + 1
. 1 + 1 + 1 + 1 + 1 + 1
-------------------------
. 1, 3, 7, 14, 21, 20
.
So row 6 lists 1, 3, 7, 14, 21, 20.
Triangle begins:
1;
1, 3;
1, 3, 5;
1, 3, 7, 9;
1, 3, 7, 12, 12;
1, 3, 7, 14, 21, 20;
1, 3, 7, 14, 24, 31, 25;
1, 3, 7, 14, 26, 40, 47, 38;
1, 3, 7, 14, 26, 43, 61, 66, 49;
1, 3, 7, 14, 26, 45, 70, 92, 93, 69:
A210953
Triangle read by rows: T(n,k) = sum of all parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.
Original entry on oeis.org
1, 0, 3, 0, 0, 5, 0, 0, 2, 9, 0, 0, 0, 3, 12, 0, 0, 0, 2, 9, 20, 0, 0, 0, 0, 3, 11, 25, 0, 0, 0, 0, 2, 9, 22, 38, 0, 0, 0, 0, 0, 3, 14, 28, 49, 0, 0, 0, 0, 0, 2, 9, 26, 44, 69, 0, 0, 0, 0, 0, 0, 3, 14, 37, 55, 87, 0, 0, 0, 0, 0, 0, 2, 9, 29, 62, 83, 123
Offset: 1
For n = 6 and k = 1..6 the 6th shell looks like this:
-------------------------
k: 1, 2, 3, 4, 5, 6
-------------------------
. 6
. 3 + 3
. 4 + 2
. 2 + 2 + 2
. 1
. 1
. 1
. 1
. 1
. 1
. 1
.
The sums of all parts in columns 1-6 are
. 0, 0, 0, 2, 9, 20, the same as the 6th row of triangle.
Triangle begins:
1;
0, 3;
0, 0, 5;
0, 0, 2, 9;
0, 0, 0, 3, 12;
0, 0, 0, 2, 9, 20;
0, 0, 0, 0, 3, 11, 25;
0, 0, 0, 0, 2, 9, 22, 38;
0, 0, 0, 0, 0, 3, 14, 28, 49;
0, 0, 0, 0, 0, 2, 9, 26, 44, 69;
0, 0, 0, 0, 0, 0, 3, 14, 37, 55, 87;
0, 0, 0, 0, 0, 0, 2, 9, 29, 62, 83, 123;
A210951
Triangle read by rows: T(n,k) = number of parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.
Original entry on oeis.org
1, 0, 2, 0, 0, 3, 0, 0, 1, 5, 0, 0, 0, 1, 7, 0, 0, 0, 1, 3, 11, 0, 0, 0, 0, 1, 3, 15, 0, 0, 0, 0, 1, 3, 6, 22, 0, 0, 0, 0, 0, 1, 4, 7, 30, 0, 0, 0, 0, 0, 1, 3, 7, 11, 42, 0, 0, 0, 0, 0, 0, 1, 4, 9, 13, 56, 0, 0, 0, 0, 0, 0, 1, 3, 8, 15, 20, 77, 0, 0, 0
Offset: 1
For n = 6 and k = 1..6 the 6th shell looks like this:
-------------------------
k: 1, 2, 3, 4, 5, 6
-------------------------
. 6
. 3 + 3
. 4 + 2
. 2 + 2 + 2
. 1
. 1
. 1
. 1
. 1
. 1
. 1
.
The total number of parts in columns 1-6 are
. 0, 0, 0, 1, 3, 11, the same as the 6th row of triangle.
Triangle begins:
1;
0, 2;
0, 0, 3;
0, 0, 1, 5;
0, 0, 0, 1, 7;
0, 0, 0, 1, 3, 11;
0, 0, 0, 0, 1, 3, 15;
0, 0, 0, 0, 1, 3, 6, 22;
0, 0, 0, 0, 0, 1, 4, 7, 30;
0, 0, 0, 0, 0, 1, 3, 7, 11, 42;
0, 0, 0, 0, 0, 0, 1, 4, 9, 13, 56;
0, 0, 0, 0, 0, 0, 1, 3, 8, 15, 20, 77;
Showing 1-3 of 3 results.