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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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,12
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)
LINKS
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<n, F be floor((n-1)/2) and S(g) be the sum of row g. Then:
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.
EXAMPLE
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
Sequence in context: A220280 A355242 A191774 * A097305 A120675 A072699
KEYWORD
nonn,tabf
AUTHOR
Bob Selcoe, Oct 04 2015
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)