A140574 Signed Pascal triangle with central coefficients set to zero.

0, -1, 1, -1, 0, -1, -1, 0, 0, 1, -1, 4, 0, 4, -1, -1, 5, 0, 0, -5, 1, -1, 6, -15, 0, -15, 6, -1, -1, 7, -21, 0, 0, 21, -7, 1, -1, 8, -28, 56, 0, 56, -28, 8, -1, -1, 9, -36, 84, 0, 0, -84, 36, -9, 1, -1, 10, -45, 120, -210, 0, -210, 120, -45, 10, -1

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Jul 05 2008

Keywords

• tabl
• less
• sign

Comments

Row sums are 0, 0, -2, 0, 6, 0, -20, 0, 70, 0, -252, ...

Examples

			0;
-1, 1;
-1, 0, -1;
-1, 0, 0, 1;
-1, 4, 0, 4, -1;
-1, 5, 0, 0, -5,1;
-1, 6, -15, 0, -15, 6, -1;
-1, 7, -21, 0, 0, 21, -7, 1;
-1, 8, -28, 56, 0,56, -28, 8, -1;
-1, 9, -36, 84, 0, 0, -84, 36, -9, 1;
-1, 10, -45, 120, -210, 0, -210, 120, -45, 10, -1;
		

Crossrefs

Cf. A007318, A130595, A091917.

Programs

Maple

A140574 := proc(n,k)
        if abs(k-n/2) < 1 and not n= 1 then
                0;
        else
        (-1)^(k+1)*binomial(n,k) ;
        end if;
end proc:
seq(seq(A140574(n,m),m=0..n),n=0..14) ; # R. J. Mathar, Nov 10 2011

Mathematica

Clear[p, f, x, n] f[x_, n_] := (-1)^ Floor[n/2]*If [Mod[n, 2] == 1, Binomial[n, Floor[n/2]]*x^( Floor[n/2]) - Binomial[n, Floor[n/2] + 1]*x^(Floor[n/2] + 1), Binomial[n, Floor[n/2]]*x^(Floor[n/2])]; p[x, 0] = 0; p[x, 1] = 1 - x; p[x_, n_] := p[x, n] = f[x, n] - (1 - x)^n; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

Formula

t(n,m) = (-1)^(m+1)*binomial(n,m) if n=1 or abs(m-n/2)>=1, otherwise t(n,m)=0.

Extensions

Adapted offset and terms to the example. - R. J. Mathar, Nov 10 2011