A158854 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial (1-x)^(1+floor(n/2))* (1+x)^floor((n-1)/2) in row n, column k.

1, 1, -1, 1, -2, 1, 1, -1, -1, 1, 1, -2, 0, 2, -1, 1, -1, -2, 2, 1, -1, 1, -2, -1, 4, -1, -2, 1, 1, -1, -3, 3, 3, -3, -1, 1, 1, -2, -2, 6, 0, -6, 2, 2, -1, 1, -1, -4, 4, 6, -6, -4, 4, 1, -1, 1, -2, -3, 8, 2, -12, 2, 8, -3, -2, 1

Offset

0,5

Comments

Row sums are zero except for n=0.

Links

Table of n, a(n) for n=0..65.

Formula

T(n,k) = T(n-2,k) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,2) = 1, T(1,1)=-1, T(2,1)=-2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Oct 25 2013

Example

1;
1, -1;
1, -2, 1;
1, -1, -1, 1;
1, -2, 0, 2, -1;
1, -1, -2, 2, 1, -1;
1, -2, -1, 4, -1, -2, 1;
1, -1, -3, 3, 3, -3, -1, 1;
1, -2, -2, 6, 0, -6, 2, 2, -1;
1, -1, -4, 4, 6, -6, -4, 4, 1, -1;
1, -2, -3, 8, 2, -12, 2, 8, -3, -2, 1;

Maple

A158854 := proc(n, k)
    (1-x)^(1+floor(n/2))*(1+x)^floor((n-1)/2) ;
    coeftayl(%, x=0, k) ;
end proc: # R. J. Mathar, Apr 08 2013

Mathematica

Clear[p, x, n, m, a];
p[x_, n_] = If[n == 0, 1, (1 - x)^(Floor[(n)/ 2] + 1)(1 + x)^(Floor[(n - 1)/2])];
Table[ExpandAll[p[x, n]], {n, 0, 10}];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A051160

Sequence in context: A294335 A194321 A194852 * A376632 A373082 A119849

Adjacent sequences: A158851 A158852 A158853 * A158855 A158856 A158857

Keyword

sign,tabl,easy

Author

Roger L. Bagula, Mar 28 2009

Status

approved