OFFSET
0,4
COMMENTS
T(n,k) is the sum of the parts of all partitions of n that contain k as a part, assuming that all partitions of n have 0 as a part: Thus, column 0 gives the sum of the parts of all partitions of n.
By definition all entries in row n>0 are divisible by n.
Row sums are 0, 2, 8, 21, 48, 95, 180, 315, 536, 873, 1390, 2145,...
The partitions of n+k that contain k as a part can be obtained by adding k to every partition of n assuming that all partitions of n have 0 as a part.
For example, the partitions of 6+k that contain k as a part are
k + 6
k + 3 + 3
k + 4 + 2
k + 2 + 2 + 2
k + 5 + 1
k + 3 + 2 + 1
k + 4 + 1 + 1
k + 2 + 2 + 1 + 1
k + 3 + 1 + 1 + 1
k + 2 + 1 + 1 + 1 + 1
k + 1 + 1 + 1 + 1 + 1 + 1
The partition number A000041(n) is also the number of partitions of m*(n+k) into parts divisible by m and that contain m*k as a part, with k>=0, m>=1, n>=0 and assuming that all partitions of n have 0 as a part.
LINKS
Robert Price, Table of n, a(n) for n = 0..5150 (First 100 rows)
FORMULA
EXAMPLE
For n=7 and k=4 there are 3 partitions of 7 that contain 4 as a part. These partitions are (4+3)=7, (4+2+1)=7 and (4+1+1+1)=7. The sum is 7+7+7 = 7*3 = 21. By other way, the partition number of 7-4 is A000041(3) = p(3)=3, then 7*3 = 21, so T(7,4) = 21.
Triangle begins with row n=0 and columns 0<=k<=n :
0,
1, 1,
4, 2, 2,
9, 6, 3, 3,
20,12,8, 4, 4,
35,25,15,10,5, 5,
66,42,30,18,12,6, 6
MAPLE
MATHEMATICA
Table[n*PartitionsP[n-k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert Price, Jun 23 2020 *)
PROG
(PARI) A182700(n, k) = n*numbpart(n-k)
CROSSREFS
Two triangles that are essentially the same as this are A027293 and A140207. - N. J. A. Sloane, Nov 28 2010
Row sums give A182704.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Nov 27 2010
STATUS
approved