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

1,2

Comments

For the construction of the diagram see A225600.

Links

Table of n, a(n) for n=1..78.

Example

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.

Sequence in context: A171852 A010287 A130270 * A090049 A115364 A016476

Adjacent sequences: A225596 A225597 A225598 * A225600 A225601 A225602

Keyword

nonn,tabl

Author

Omar E. Pol, Aug 02 2013

Status

approved