A137296 Antidiagonal triangular matrices of factorials as the example: M(3)={{0, 0, 1}, {0, 1, 2}, {1, 2, 6}}; the matrices are used to get characteristic polynomials and the triangular sequence is the coefficients of those characteristic polynomials.

1, 0, -1, -1, -2, 1, -1, -1, 7, -1, 1, 10, 6, -26, 1, 1, 45, -36, -224, 127, -1, -1, -306, -957, 1172, 3709, -746, 1, -1, -2357, -14379, 37397, 67621, -106887, 5167, -1, 1, 20690, 803492, 1325962, -4532586, -6300786, 4147412, -41066, 1, 1, 201733, 49001186, 51278654, -546386064, -208428708, 1389178142

Offset

1,5

Comments

This set of matrices was inspired by the Cornelius-Schultz triangular n! matrices and the Steinbach like ones antidiagonal matrix:

The row sums are {1, -1, -2, 4, -8, -88, 2872, -13440, -4576880, 511742560, 19591271488}.

Links

Table of n, a(n) for n=1..52.

Formula

M(n)->If greater than or equal to the middle antidiagonal, the antidiagonals are counted n->n!. Main antidiagonal is 1; second antidiagonal is 2!=1; third antidiagonal is 3!=6, etc.

Example

{1},
{0, -1},
{-1, -2, 1},
{-1, -1, 7, -1},
{1, 10, 6, -26, 1},
{1, 45, -36, -224, 127, -1},
{-1, -306, -957, 1172, 3709, -746, 1},
{-1, -2357, -14379, 37397, 67621, -106887,5167, -1},
{1, 20690,803492, 1325962, -4532586, -6300786, 4147412, -41066, 1},
{1, 201733, 49001186, 51278654, -546386064, -208428708, 1389178142, -223470430, 368047, -1},
{-1, -2159026, -4071243531, -42916351880, -16447372250, 291803389596, 168497207674, -392959445624, 15690916395, -3669866, 1}

Mathematica

T[d_] := Table[Table[If[n == m && n == 1, 0, If[n == m && (n + m) > d, ((m + n) - d)!, If[Abs[n - m] >= 0 && (n + m) > d, ((m + n) - d)!, 0]]], {n, 1, d}], {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[T[d], x], x], \ {d, 1, 10}]]; Flatten[a]

Crossrefs

Sequence in context: A049270 A025269 A265656 * A329291 A294105 A101124

Adjacent sequences: A137293 A137294 A137295 * A137297 A137298 A137299

Keyword

uned,sign

Author

Roger L. Bagula, Mar 15 2008

Status

approved