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A182700 Triangle T(n,k) = n*A000041(n-k), 0<=k<=n, read by rows. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A136202 A075418 A199221

Adjacent sequences:  A182697 A182698 A182699 * A182701 A182702 A182703

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Nov 27 2010

STATUS

approved

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Last modified August 14 09:14 EDT 2020. Contains 336480 sequences. (Running on oeis4.)