A182700 Triangle T(n,k) = n*A000041(n-k), 0<=k<=n, read by rows.

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, 105, 77, 49, 35, 21, 14, 7, 7, 176, 120, 88, 56, 40, 24, 16, 8, 8, 270, 198, 135, 99, 63, 45, 27, 18, 9, 9, 420, 300, 220, 150, 110, 70, 50, 30, 20, 10, 10, 616, 462, 330, 242, 165, 121, 77, 55, 33

Offset: 0

Author

Omar E. Pol, Nov 27 2010

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• Robert Price, Table of n, a(n) for n = 0..5150 (First 100 rows)

Crossrefs

Cf. A000041, A027293, A135010, A138121.

Two triangles that are essentially the same as this are A027293 and A140207. - N. J. A. Sloane, Nov 28 2010

Row sums give A182704.

Programs

Maple

A182700 := proc(n,k) n*combinat[numbpart](n-k) ; end proc:
seq(seq(A182700(n,k),k=0..n),n=0..15) ;

Mathematica

Table[n*PartitionsP[n-k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert Price, Jun 23 2020 *)

PARI

A182700(n,k) = n*numbpart(n-k)

Formula

T(n,0) = A066186(n).

T(n,k) = A182701(n,k), n>=1 and k>=1.

T(n,n) = n = min { T(n,k); 0<=k<=n }.