A124031 - OEIS

A124031

Triangle read by rows: T(0, 0) = -1. T(n, k) = [x^k] det(M - x*I), where M is an n X n matrix defined by M(i, j) = (-1)^i if i = j, M(i, j) = -1 if |i-j| = 1, and M(i, j) = 0 otherwise.

-1, -1, -1, -2, 0, 1, 3, 3, -1, -1, 5, 0, -5, 0, 1, -8, -8, 6, 6, -1, -1, -13, 0, 19, 0, -8, 0, 1, 21, 21, -25, -25, 9, 9, -1, -1, 34, 0, -65, 0, 42, 0, -11, 0, 1, -55, -55, 90, 90, -51, -51, 12, 12, -1, -1, -89, 0, 210, 0, -183, 0, 74, 0, -14, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

Triangle starts:

{-1},

{-1, -1},

{-2, 0, 1},

{3, 3, -1, -1},

{5, 0, -5, 0, 1},

{-8, -8, 6, 6, -1, -1},

{-13, 0, 19, 0, -8, 0, 1},

{21, 21, -25, -25, 9, 9, -1, -1},

{34, 0, -65, 0, 42, 0, -11, 0, 1},

{-55, -55, 90, 90, -51, -51, 12, 12, -1, -1},

{-89, 0, 210, 0, -183, 0, 74, 0, -14, 0, 1}, ...

MATHEMATICA

T[n_, m_, d_] := If[n == m, (-1)^n, If[n == m - 1 || n == m + 1, -1, 0]];

M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];

Flatten@ Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 10}] ]

PROG

(PARI) T(n, k) = {if(n==0&&k==0, return(-1)); my(M=matrix(n, n, i, j, if(i==j, (-1)^i, if(abs(i-j)==1, -1, 0)))); polcoef(matdet(M-x), k)} \\ Jason Yuen, Apr 07 2025

CROSSREFS

Sequence in context: A099390 A297477 A370030 * A368514 A289229 A263097

Adjacent sequences: A124028 A124029 A124030 * A124032 A124033 A124034

KEYWORD

tabl,less,sign

AUTHOR

Roger L. Bagula and Gary W. Adamson, Nov 01 2006

EXTENSIONS

Edited and new name from Jason Yuen, Apr 07 2025

STATUS

approved