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A140682 Triangle T(n,k) = gcd(n,k)-binomial(n,k), 0<=k<=n.

%I A140682 #8 Jan 17 2013 11:09:38
%S A140682 -1,0,0,1,-1,1,2,-2,-2,2,3,-3,-4,-3,3,4,-4,-9,-9,-4,4,5,-5,-13,-17,
%T A140682 -13,-5,5,6,-6,-20,-34,-34,-20,-6,6,7,-7,-26,-55,-66,-55,-26,-7,7,8,
%U A140682 -8,-35,-81,-125,-125,-81,-35,-8,8,9,-9,-43,-119,-208,-247,-208,-119,-43,-9,9
%N A140682 Triangle T(n,k) = gcd(n,k)-binomial(n,k), 0<=k<=n.
%C A140682 Row sums are -1, 0, 1, 0, -4, -18, -43, -108, -228, -482, -987...
%F A140682 T(n,k) = T(n,n-k).
%F A140682 T(n,k) = A109004(n,k)-A007318(n,k). - _R. J. Mathar_, Jan 17 2013
%e A140682 -1;
%e A140682 0, 0;
%e A140682 1, -1, 1;
%e A140682 2, -2, -2, 2;
%e A140682 3, -3, -4, -3, 3;
%e A140682 4, -4, -9, -9, -4, 4;
%e A140682 5, -5, -13, -17, -13, -5, 5;
%e A140682 6, -6, -20, -34, -34, -20, -6, 6;
%e A140682 7, -7, -26, -55, -66, -55, -26, -7, 7;
%e A140682 8, -8, -35, -81, -125, -125, -81, -35, -8, 8;
%e A140682 9, -9, -43, -119, -208, -247, -208, -119, -43, -9, 9;
%p A140682 A140682 := proc(n,k)
%p A140682     igcd(n,k)-binomial(n,k) ;
%p A140682 end proc: # _R. J. Mathar_, Jan 17 2013
%t A140682 Clear[p, x, n] p[x_, n_] = Sum[(GCD[n, i] - Binomial[n, i])*x^i, {i, 0, n}]; Table[ExpandAll[p[x, n]], {n, 1, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 1, 10}]; Flatten[a]
%Y A140682 Cf. A109004.
%K A140682 tabl,easy,sign
%O A140682 0,7
%A A140682 _Roger L. Bagula_ and _Gary W. Adamson_, Jul 11 2008
%E A140682 New name, editing, and missing leading terms added. - _R. J. Mathar_, Jan 17 2013