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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.

%I A158854 #11 Oct 27 2013 04:40:31
%S A158854 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,
%T A158854 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,
%U A158854 -2,-3,8,2,-12,2,8,-3,-2,1
%N 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.
%C A158854 Row sums are zero except for n=0.
%F A158854 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
%e A158854 1;
%e A158854 1, -1;
%e A158854 1, -2, 1;
%e A158854 1, -1, -1, 1;
%e A158854 1, -2, 0, 2, -1;
%e A158854 1, -1, -2, 2, 1, -1;
%e A158854 1, -2, -1, 4, -1, -2, 1;
%e A158854 1, -1, -3, 3, 3, -3, -1, 1;
%e A158854 1, -2, -2, 6, 0, -6, 2, 2, -1;
%e A158854 1, -1, -4, 4, 6, -6, -4, 4, 1, -1;
%e A158854 1, -2, -3, 8, 2, -12, 2, 8, -3, -2, 1;
%p A158854 A158854 := proc(n,k)
%p A158854     (1-x)^(1+floor(n/2))*(1+x)^floor((n-1)/2) ;
%p A158854     coeftayl(%,x=0,k) ;
%p A158854 end proc: # _R. J. Mathar_, Apr 08 2013
%t A158854 Clear[p, x, n, m, a];
%t A158854 p[x_, n_] = If[n == 0, 1, (1 - x)^(Floor[(n)/ 2] + 1)(1 + x)^(Floor[(n - 1)/2])];
%t A158854 Table[ExpandAll[p[x, n]], {n, 0, 10}];
%t A158854 Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
%t A158854 Flatten[%]
%Y A158854 Cf. A051160
%K A158854 sign,tabl,easy
%O A158854 0,5
%A A158854 _Roger L. Bagula_, Mar 28 2009