A076756 Triangle of coefficients of characteristic polynomial of M_n, the n X n matrix M_(i,j) = min(i,j).

1, -1, 1, 1, -3, 1, -1, 5, -6, 1, 1, -7, 15, -10, 1, -1, 9, -28, 35, -15, 1, 1, -11, 45, -84, 70, -21, 1, -1, 13, -66, 165, -210, 126, -28, 1, 1, -15, 91, -286, 495, -462, 210, -36, 1, -1, 17, -120, 455, -1001, 1287, -924, 330, -45, 1, 1, -19, 153, -680, 1820, -3003, 3003, -1716, 495, -55, 1

Offset: 0

Author

Benoit Cloitre, Nov 09 2002

Keywords

• sign
• tabl

Comments

The characteristic polynomial of M_n seems to be p(n,x) = (-1)^n * sum_{i=0..n} (-x)^i * binomial(2n-i, i). - Enrique Pérez Herrero, Jan 29 2013

Examples

			Triangle begins:
   1;
  -1,   1;
   1,  -3,   1;
  -1,   5,  -6,   1;
   1,  -7,  15, -10,    1;
  -1,   9, -28,  35,  -15,   1;
   1, -11,  45, -84,   70, -21,   1;
  -1,  13, -66, 165, -210, 126, -28,  1;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened (rows 0..60 from Enrique Pérez Herrero)
• J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444. See Eq. 2.27. - _N. J. A. Sloane_, Mar 14 2014

Crossrefs

Absolute values are A054142.

Cf. A203989.

Programs

Maple

T:=(n, k)-> binomial(2*n-k, k)*(-1)^(n+k):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Feb 01 2013

Mathematica

T[n_, k_] := Binomial[2*n - k, k]*(-1)^(n + k); Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 12 2015, after Alois P. Heinz *)

Extensions

Offset corrected by Alois P. Heinz, Feb 01 2013