A066170 Triangle read by rows: T(n,k) = (-1)^n*(-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k), 0 <= k <= n, n >= 0.

1, -1, 1, 1, -1, -1, -1, 2, 1, -1, 1, -2, -3, 1, 1, -1, 3, 3, -4, -1, 1, 1, -3, -6, 4, 5, -1, -1, -1, 4, 6, -10, -5, 6, 1, -1, 1, -4, -10, 10, 15, -6, -7, 1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, 1, -5, -15, 20, 35, -21, -28, 8, 9, -1, -1, -1, 6, 15, -35, -35, 56, 28, -36, -9, 10, 1, -1, 1, -6, -21, 35, 70, -56, -84, 36, 45, -10, -11

Offset: 0

Author

Floor van Lamoen, Dec 14 2001

Keywords

• sign
• easy
• tabl

Comments

The original name of this sequence was: Triangle giving coefficients of characteristic function of n X n matrix in which the left upper half and the antidiagonal are filled with 1's and the right lower half is filled with 0's. As was pointed out by L. Edson Jeffery this is only correct if we multiply each triangle row by (-1)^n. For the straightforward version of the coefficients of the characteristic polynomials see A187660. - Johannes W. Meijer, Aug 08 2011

Examples

			The table begins {1}; {-1, 1}; {1, -1, -1}; {-1, 2, 1, -1}; ...
The characteristic function of
( 1 1 1 )
( 1 1 0 )
( 1 0 0 )
is f(x) = x^3 - 2x^2 - x + 1, so the 3rd row is (-1)^3 times the f(x) coefficients, i.e., {-1; 2; 1; -1}.
		

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)

Links

• Indranil Ghosh, Rows 0..125, flattened
• Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.
• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Crossrefs

Cf. A007700, A059455, A065941. For another version see A030111.

Programs

Maple

A066170 := proc(n,k): (-1)^n*(-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k) end: seq(seq(A066170(n,k),k=0..n), n=0..11); // Johannes W. Meijer, Aug 08 2011

Mathematica

Flatten[Table[(-1)^n*(-1)^Floor[3*k/2]*Binomial[Floor[(n+k)/2],k],{n,0,12}, {k,0,n}]] (* Indranil Ghosh, Feb 19 2017 *)

Formula

From L. Edson Jeffery, Mar 23 2011: (Start)

T(n,k) = (-1)^n*(-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k);

T(n,k) = (-1)^n*A187660(n,k). (End)

From Johannes W. Meijer, Aug 08 2011: (Start)

abs(T(n,k)) = A046854(n,k) = abs(A108299(n,n-k))

abs(T(n,n-k)) = A065941(n,k). (End)

Extensions

More terms from Vladeta Jovovic, Jan 02 2002

Corrected and edited by Johannes W. Meijer, Aug 08 2011