OFFSET
1,3
COMMENTS
For the definition of "region" see A206437.
T(n,k) is also the number of parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).
EXAMPLE
For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the total number of parts is 3 + 1 = 4, so T(5,3) = 4.
.
. Diagram Illustration of parts ending 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: 1 3 4 5 7
.
Triangle begins:
1;
1, 2;
1, 2, 3;
1, 3, 3, 5;
1, 3, 4, 5, 7;
1, 4, 5, 7, 7, 11;
1, 4, 6, 8, 9, 11, 15;
1, 5, 7, 11, 10, 15, 15, 22;
1, 5, 9, 12, 13, 17, 19, 22, 30;
1, 6, 10, 16, 15, 22, 21, 29, 30, 42;
1, 6, 12, 18, 19, 25, 26, 32, 38, 42, 56;
1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77;
CROSSREFS
Column 1 is A000012. Column 2 are the numbers => 2 of A008619. Row sums give A006128, n>=1. Right border gives A000041, n>=1. Second right border gives A000041, n>=1.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Aug 02 2013
STATUS
approved