A145661 Triangle T(n,k) = (-1)^k * A119258(n,k) read by rows, 0 <= k <= n.

1, 1, -1, 1, -3, 1, 1, -5, 7, -1, 1, -7, 17, -15, 1, 1, -9, 31, -49, 31, -1, 1, -11, 49, -111, 129, -63, 1, 1, -13, 71, -209, 351, -321, 127, -1, 1, -15, 97, -351, 769, -1023, 769, -255, 1, 1, -17, 127, -545, 1471, -2561, 2815, -1793, 511, -1, 1, -19, 161, -799, 2561

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Mar 16 2009

Keywords

• tabl
• easy
• sign

Comments

Row sums are (-1)^(n+1)*(n-1) for n >= 1.

A145661, A119258 and A118801 are all essentially the same (see the Shattuck and Waldhauser paper). - Tamas Waldhauser, Jul 25 2011

Examples

			Triangle begins
  1;
  1,  -1;
  1,  -3,   1;
  1,  -5,   7,   -1;
  1,  -7,  17,  -15,    1;
  1,  -9,  31,  -49,   31,    -1;
  1, -11,  49, -111,  129,   -63,    1;
  1, -13,  71, -209,  351,  -321,  127,    -1;
  1, -15,  97, -351,  769, -1023,  769,  -255,    1;
  1, -17, 127, -545, 1471, -2561, 2815, -1793,  511,    -1;
  1, -19, 161, -799, 2561, -5503, 7937, -7423, 4097, -1023, 1;
		

Links

• J.-F. Chamayou, A Random Difference Equation with Dufresne Variables Revisited, arXiv preprint arXiv:1410.1708 [math.PR], 2014. See Table in Section XII.
• M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Quart., 48 (2010), 290-297.

Crossrefs

Cf. A135233, A112857.

A193844 is an essentially identical triangle.

Programs

Maple

A119258 := proc(n,k)
        if k=0 or k = n then
                1;
        elif k<0 or k> n then
                0;
        else
                2*procname(n-1,k-1)+procname(n-1,k) ;
        end if;
end proc:
seq(seq(A119258(n,k),k=0..n),n=0..10) ;
A145661 := proc(n,k)
        (-1)^k*A119258(n,k) ;
end proc: # R. J. Mathar, Oct 21 2011

Mathematica

Clear[M, T, d, a, x, a0];
T[n_, m_, d_] := If[ m == n + 1, 1, If[n == d, 1, 0]];
M[d_] := MatrixPower[Table[T[n, m, d], {n, 1, d}, {m, 1, d}], d];
Table[M[d], {d, 1, 10}];
Table[Det[M[d]], {d, 1, 10}];
Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
a = Join[{{1}}, Table[CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x], {n, 1, 10}]];
Flatten[a]
Join[{1}, Table[Apply[ Plus, CoefficientList[Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];