A122944 Triangle read by rows: Row n is the expansion of the characteristic polynomial of an n X n submatrix of A131218 (0 <= i,j < n).

1, 1, -1, -1, -1, 1, 0, 2, 1, -1, 1, -1, -4, -1, 1, 0, -2, 2, 6, 1, -1, 0, 0, 4, -2, -7, -1, 1, 0, 2, -1, -9, 3, 9, 1, -1, 1, 1, -13, 8, 20, -8, -13, -1, 1, 0, -2, -2, 24, -15, -31, 13, 17, 1, -1, 0, 0, 4, 4, -40, 20, 44, -14, -19, -1, 1, 0, 0, 0, -8, -4, 56, -24, -54, 14, 20, 1, -1, 0, 0, 0, 0, 16, 8, -88, 30, 71, -15, -22, -1, 1, 0, 0, 0, 16, 8

Offset

1,8

Links

Table of n, a(n) for n=1..96.

Roger Bagula and Gary Adamson, Pascal's Triangle in Gray Code: its Hadamard and IFS.

Example

Triangle:
n\k| 0  1   2   3   4   5   6   7   8   9 10 11
---+-------------------------------------------
  0| 1,
  1|-1, 1
  2|-1,-1,  1
  3| 0,-2, -1,  1
  4| 1,-1, -4, -1,  1
  5| 0, 2, -2, -6, -1,  1
  6| 0, 0,  4, -2, -7, -1,  1
  7| 0,-2,  1,  9, -3, -9, -1,  1
  8| 1, 1,-13,  8, 20, -8,-13, -1,  1
  9| 0, 2,  2,-24, 15, 31,-13,-17, -1,  1
 10| 0, 0,  4,  4,-40, 20, 44,-14,-19, -1, 1
 11| 0, 0,  0,  8,  4,-56, 24, 54,-14,-20,-1, 1
Polynomials:
  1,
 -1 + x,
 -1 - x + x^2,
    -2x - x^2 + x^3,
  1 - x -4x^2 - x^3 + x^4,
     2x -2x^2 -6x^3 - x^4 + x^5,
         4x^2 -2x^3 -7x^4 - x^5 + x^6,
    -2x + x^2 +9x^3 -3x^4 -9x^5 - x^6 + x^7

Mathematica

An[n_]:=CoefficientList[(-1)^n*CharacteristicPolynomial[Table[If[BitAnd[BitXor[i, Floor[i/2]], BitXor[j, Floor[j/2]]]==0, 1, 0], {i, 0, n-1}, {j, 0, n-1}], x], x]
a=Join[{{1}}, Table[An[d], {d, 1, 20}]]
Flatten[a]
RowSum=Table[Apply[Plus, Abs[a[[n]]]], {n, 1, Length[a]}] (* reduced by Natalia L. Skirrow, Nov 17 2025 *)

Prog

(SageMath)
def A122944(n: int) -> list[int]:
    def gray_code(x: int) -> int: return x^^(x >> 1)
    data = [[int(not (gray_code(i) & gray_code(j)))
           for j in range(n)] for i in range(n)]
    poly = matrix(ZZ, data).charpoly()
    return [int(c) for c in poly.list()]
for n in range(12): print(A122944(n))  # Peter Luschny, Nov 18 2025

Crossrefs

Cf. A121801.

Sequence in context: A118210 A061399 A161856 * A107688 A376267 A060097

Adjacent sequences: A122941 A122942 A122943 * A122945 A122946 A122947

Keyword

tabl,sign

Author

Roger L. Bagula and Gary W. Adamson, Oct 24 2006

Extensions

Name updated and sign convention changed (multiplied by (-1)^n to keep T(n,n) positive) by Natalia L. Skirrow, Nov 17 2025

Status

approved