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

0,4

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

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 }.

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

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 *)

Prog

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

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.

Sequence in context: A079184 A095800 A055630 * A369408 A136202 A075418

Adjacent sequences: A182697 A182698 A182699 * A182701 A182702 A182703

Keyword

nonn,tabl

Author

Omar E. Pol, Nov 27 2010

Status

approved