A225599 Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n.

1, 3, 1, 6, 1, 2, 12, 1, 4, 3, 20, 1, 4, 5, 5, 35, 1, 6, 8, 9, 7, 54, 1, 6, 10, 12, 11, 11, 86, 1, 8, 13, 20, 14, 19, 15, 128, 1, 8, 18, 23, 22, 25, 23, 22, 192, 1, 10, 21, 34, 30, 37, 29, 36, 30, 275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42, 399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56

Offset: 1

Author

Omar E. Pol, Aug 02 2013

Keywords

• nonn
• tabl

Comments

For the construction of the diagram see A225600.

Examples

			For n = 5 and k = 3 the diagram of regions of the set of partitions of 5 contains three parts that start in the third column: two parts of size 1 and one part of size 2, therefore the sum of all parts that start in column 3 is 1 + 1 + 2 = 4, so T(5,3) = 4.
.
.                       Illustration of the parts
.    Diagram             that start in column k:
.    for n=5       k=1          k=2  k=3    k=4    k=5
.   _ _ _ _ _       _ _ _ _ _
.  |_ _ _    |     |_ _ _ _ _|               _ _
.  |_ _ _|_  |     |_ _ _|_                 |_ _|   _
.  |_ _    | |     |_ _ _ _|          _ _          |_|
.  |_ _|_  | |     |_ _|_            |_ _|   _     |_|
.  |_ _  | | |     |_ _ _|            _     |_|    |_|
.  |_  | | | |     |_ _|         _   |_|    |_|    |_|
.  |_|_|_|_|_|     |_|          |_|  |_|    |_|    |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  20           1    4      5      5
.
Triangle begins:
1;
3,   1;
6,   1,  2;
12,  1,  4,  3;
20,  1,  4,  5,  5;
35,  1,  6,  8,  9,  7;
54,  1,  6, 10, 12, 11, 11;
86,  1,  8, 13, 20, 14, 19, 15;
128, 1,  8, 18, 23, 22, 25, 23, 22;
192, 1, 10, 21, 34, 30, 37, 29, 36, 30;
275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42;
399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56;
		

Crossrefs

Column 1-2: A006128, A000012. Row sums give A066186. Right border gives A000041.

Cf. A135010, A141285, A176572, A186114, A186412, A187219, A193870, A194446, A206437, A211978, A225598, A225600, A225610.