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}.

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

Offset: 1

Author

Wolfdieter Lang, Feb 15 2021

Keywords

• sign
• tabl

Comments

The row sums are given in A341417.

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) = A257628(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.

Examples

			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

Cf. A000045, A001318, A008284, -A010815, A257628, A341417, A341418.

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.