A076756 - OEIS

%I #39 Oct 27 2023 21:41:11

%S 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,

%T 70,-21,1,-1,13,-66,165,-210,126,-28,1,1,-15,91,-286,495,-462,210,-36,

%U 1,-1,17,-120,455,-1001,1287,-924,330,-45,1,1,-19,153,-680,1820,-3003,3003,-1716,495,-55,1

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

%C 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

%H Alois P. Heinz, <a href="/A076756/b076756.txt">Rows n = 0..140, flattened</a> (rows 0..60 from Enrique Pérez Herrero)

%H J. L. Jacobsen, and J. Salas, <a href="http://arxiv.org/abs/cond-mat/0407444">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions</a>, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444. See Eq. 2.27. - _N. J. A. Sloane_, Mar 14 2014

%e Triangle begins:

%e    1;

%e   -1,   1;

%e    1,  -3,   1;

%e   -1,   5,  -6,   1;

%e    1,  -7,  15, -10,    1;

%e   -1,   9, -28,  35,  -15,   1;

%e    1, -11,  45, -84,   70, -21,   1;

%e   -1,  13, -66, 165, -210, 126, -28,  1;

%p T:=(n, k)-> binomial(2*n-k, k)*(-1)^(n+k):

%p seq(seq(T(n, k), k=0..n), n=0..14);  # _Alois P. Heinz_, Feb 01 2013

%t 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_ *)

%Y Absolute values are A054142.

%Y Cf. A203989.

%K sign,tabl

%O 0,5

%A _Benoit Cloitre_, Nov 09 2002

%E Offset corrected by _Alois P. Heinz_, Feb 01 2013