A262885 Irregular triangle T(n,k) read by rows: T(n,k) = number of partitions of n into at least two distinct parts, where the largest part is n-k.

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 1, 2, 2, 3, 3, 2, 1, 1, 2, 2, 3, 4, 3, 1, 1, 1, 2, 2, 3, 4, 4, 3, 1, 1, 1, 2, 2, 3, 4, 5, 4, 3, 1, 1, 1, 2, 2, 3, 4, 5, 5, 5, 3, 1, 1, 2, 2, 3, 4, 5, 6, 6, 5, 2, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 5, 2

Offset: 1

Author

Bob Selcoe, Oct 04 2015

Keywords

• nonn
• tabf

Comments

Alternate name: T(n,k) = the number of ways that at least two distinct positive integers sum to n, where the largest of these integers is n-k.

Row sums = A111133(n).

Row sums {k <= floor((n-1)/2)} = A026906(n)

Row sums {k > floor((n-1)/2)} = A258259(n)

Examples

			Triangle starts T(1,1):
n/k  1 2 3 4 5 6 7 8 9 10 11 12 13 14
1    0
2    0
3    1
4    1
5    1 1
6    1 1 1
7    1 1 2
8    1 1 2 1
9    1 1 2 2 1
10   1 1 2 2 2 1
11   1 1 2 2 3 2
12   1 1 2 2 3 3 2
13   1 1 2 2 3 4 3 1
14   1 1 2 2 3 4 4 3 1
15   1 1 2 2 3 4 5 4 3 1
16   1 1 2 2 3 4 5 5 5 3
17   1 1 2 2 3 4 5 6 6 5  2
18   1 1 2 2 3 4 5 6 7 7  5  2
19   1 1 2 2 3 4 5 6 8 8  7  5  1
20   1 1 2 2 3 4 5 6 8 9  9  8  4  1
T(15,8) = 4: the four partitions of 15 into at least two distinct parts with largest part 15-8 = 7 are  {7,6,2}; {7,5,3}; {7,5,2,1} and {7,4,3,1}.
T(14,k) for k=1..F, with F = floor(13/2) = 6: T(14,1) = 0+1 = 1; T(14,2) = 0+1 = 1; T(14,3) = 1+1 = 2; T(14,4) = 1+1 = 2; T(14,5) = 2+1 = 3; T(14,6) = 3+1 = 4.
T(14,k) for k>6: T(14,7) = T(7,1)+T(7,2)+T(7,3) = 1+1+2 = 4; T(14,8) = T(8,3)+T(8,4) = 2+1 = 3; T(14,9) = T(9,5) = 1.
		

Crossrefs

Cf. A000009, A026906, A111133, A258259.

Formula

Given T(1,1) = T(2,1) = 0, to find row n>=3: Let k" be the maximum value of k in row g

T(n,k) = S(g)+1 g=k when g<=F (equivalent to A000009(g));

T(n,k) = Sum_{j=2*(g-F)-1..k"} T(g,j) g=k when g>F, 2*(g-F)-1 <= k" and n is even;

T(n,k) = Sum_{j=2*(g-F)..k"} T(g,j) g=k when g>F, 2*(g-F) <= k" and n is odd.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.