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A140574
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Signed Pascal triangle with central coefficients set to zero.
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1
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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
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OFFSET
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0,12
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COMMENTS
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Row sums are 0, 0, -2, 0, 6, 0, -20, 0, 70, 0, -252, ...
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LINKS
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FORMULA
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t(n,m) = (-1)^(m+1)*binomial(n,m) if n=1 or abs(m-n/2)>=1, otherwise t(n,m)=0.
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EXAMPLE
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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;
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MAPLE
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if abs(k-n/2) < 1 and not n= 1 then
0;
else
(-1)^(k+1)*binomial(n, k) ;
end if;
end proc:
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MATHEMATICA
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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[%]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Adapted offset and terms to the example. - R. J. Mathar, Nov 10 2011
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STATUS
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approved
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