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A140574 Signed Pascal triangle with central coefficients set to zero.

%I A140574 #10 Oct 03 2015 23:52:48
%S A140574 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,
%T A140574 -1,-1,7,-21,0,0,21,-7,1,-1,8,-28,56,0,56,-28,8,-1,-1,9,-36,84,0,0,
%U A140574 -84,36,-9,1,-1,10,-45,120,-210,0,-210,120,-45,10,-1
%N A140574 Signed Pascal triangle with central coefficients set to zero.
%C A140574 Row sums are 0, 0, -2, 0, 6, 0, -20, 0, 70, 0, -252, ...
%F A140574 t(n,m) = (-1)^(m+1)*binomial(n,m) if n=1 or abs(m-n/2)>=1, otherwise t(n,m)=0.
%e A140574 0;
%e A140574 -1, 1;
%e A140574 -1, 0, -1;
%e A140574 -1, 0, 0, 1;
%e A140574 -1, 4, 0, 4, -1;
%e A140574 -1, 5, 0, 0, -5,1;
%e A140574 -1, 6, -15, 0, -15, 6, -1;
%e A140574 -1, 7, -21, 0, 0, 21, -7, 1;
%e A140574 -1, 8, -28, 56, 0,56, -28, 8, -1;
%e A140574 -1, 9, -36, 84, 0, 0, -84, 36, -9, 1;
%e A140574 -1, 10, -45, 120, -210, 0, -210, 120, -45, 10, -1;
%p A140574 A140574 := proc(n,k)
%p A140574         if abs(k-n/2) < 1 and not n= 1 then
%p A140574                 0;
%p A140574         else
%p A140574         (-1)^(k+1)*binomial(n,k) ;
%p A140574         end if;
%p A140574 end proc:
%p A140574 seq(seq(A140574(n,m),m=0..n),n=0..14) ; # _R. J. Mathar_, Nov 10 2011
%t A140574 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[%]
%Y A140574 Cf. A007318, A130595, A091917.
%K A140574 tabl,less,sign
%O A140574 0,12
%A A140574 _Roger L. Bagula_ and _Gary W. Adamson_, Jul 05 2008
%E A140574 Adapted offset and terms to the example. - _R. J. Mathar_, Nov 10 2011