|
| |
|
|
A339885
|
|
Triangle read by rows: T(n, m) gives the sum of the weights of weighted partitions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.
|
|
2
|
|
|
|
1, 1, 1, 0, 1, 1, 0, 1, 1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 0, -1, -1, 0, 1, 1, 1, 1, 0, -1, -2, -1, 0, 1, 1, 1, 1, 0, 1, -1, -2, 0, 0, 1, 1, 1, 1, 0, 0, 0, -2, -2, 0, 0, 1, 1, 1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,39
|
|
|
COMMENTS
|
One could add a row n=0 and the column (1,repeat(0)) including the empty partition with no parts, and number of parts m = 0. The weight w(0) = -1.
The weight from {-1, 0, +1} of a positive number n is w(n) = 0 if n is not an element of the generalized pentagonal numbers {Pent(k) = A001318(k)}_{k>=1}, and if n = Pent(k) then w(n) = (-1)^(ceiling(Pent(k)/2)+1). The sequence
{(n, w(n))}_{n>=1} begins: {(1,+1), (2,+1), (3,0), (4,0), (5,-1), (6,0), (7,-1), ...}. One can also use w(0) = -1. w(n) = -A010815(n), for n >= 0. For n >= 1 w(n) = A257028(n) also.
The weight of a partition is the product of the weights of its parts.
For the triangle giving the sum of the weights of weighted compositions of n with m parts from the generalized pentagonal numbers see A341418.
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n, m) = Sum_{j=1..p(n,m)} w(Part(n, m, j)), where p(n, m) = A008284(n, m), and the ternary weight of the j-th partition of n with m parts Part(n,m,j), in Abramowitz-Stegun order, is defined as the product of the weights of the parts, by w(n) = -A010815(n), for n >= 1 and m = 1, 2, ..., n.
|
|
|
EXAMPLE
|
The triangle T(n, m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... A341417
----------------------------------------------------------------------------
1: 1 1
2: 1 1 2
3: 0 1 1 2
4: 0 1 1 1 3
5; -1 0 1 1 1 2
6: 0 -1 1 1 1 1 3
7: -1 -1 -1 1 1 1 1 1
8: 0 -1 -1 0 1 1 1 1 2
9: 0 -1 -2 -1 0 1 1 1 1 0
10: 0 1 -1 -2 0 0 1 1 1 1 2
11: 0 0 0 -2 -2 0 0 1 1 1 1 0
12: 1 1 1 0 -2 -1 0 0 1 1 1 1 4
13: 0 1 1 0 -1 -2 -1 0 0 1 1 1 1 2
14: 0 2 2 2 0 -1 -1 -1 0 0 1 1 1 1 7
15: 1 0 1 2 1 -1 -1 -1 -1 0 0 1 1 1 1 5
16: 0 1 2 2 3 1 -1 0 -1 -1 0 0 1 1 1 1 10
17: 0 0 0 1 2 2 0 -1 0 -1 -1 0 0 1 1 1 1 6
18: 0 0 0 2 2 3 2 0 0 0 -1 -1 0 0 1 1 1 1 11
19: 0 -1 -1 -1 1 2 2 1 0 0 0 -1 -1 0 0 1 1 1 1 5
20: 0 -1 -1 0 1 2 3 2 1 1 0 0 -1 -1 0 0 1 1 1 1 10
...
n = 5: (Partition; weight w) with | separating same m numbers (in Abramowitz -Stegun order):
(5;-1) | (1,4;0), (2,3;0) | (1^2,3;0), (1,2^2;1) | (1^3,2;1) | (1^5;1), hence row n=5 is [-1, 0, 1, 1, 1] from the sum of same m weights.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
