%I A110555
%S A110555 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,4,1,0,1,5,10,10,5,1,0,1,6,15,20,15,
%T A110555 6,1,0,1,7,21,35,35,21,7,1,0,1,8,28,56,70,56,28,8,1,0,1,9,36,84,126,126,
%U A110555 84,36,9,1,0,1,10,45,120,210,252,210,120
%V A110555 1,1,0,1,-1,0,1,-2,1,0,1,-3,3,-1,0,1,-4,6,-4,1,0,1,-5,10,-10,5,-1,0,1,
               -6,15,-20,15,-6,
%W A110555 1,0,1,-7,21,-35,35,-21,7,-1,0,1,-8,28,-56,70,-56,28,-8,1,0,1,-9,36,-84,
               126,-126,84,
%X A110555 -36,9,-1,0,1,-10,45,-120,210,-252,210,-120
%N A110555 Triangle of partial sums of alternating binomial coefficients: T(n,k) 
               = Sum(binomial(n,k)*(-1)^k: 0<=k<=n).
%C A110555 T(n,0)=1, T(n,n)=0^n, T(n,k)=-T(n-1,k-1)+T(n-1,k), 0<k<n;
%C A110555 T(n,n-k-1) = -T(n,k), 0<k<n;
%C A110555 A071919(n,k) = abs(T(n,k)), T(n,k) = A071919(n,k)*(-1)^k;
%C A110555 row sums give A000007; central terms give A110556;
%C A110555 T(n,1) = -n + 1 for n>0;
%C A110555 T(n,2) = A000217(n-2) for n>1;
%C A110555 T(n,3) = -A000292(n-4) for n>2;
%C A110555 T(n,4) = A000332(n-1) for n>3;
%C A110555 T(n,5) = -A000389(n-1) for n>5;
%C A110555 T(n,6) = A000579(n-1) for n>6;
%C A110555 T(n,7) = -A000580(n-1) for n>7;
%C A110555 T(n,8) = A000581(n-1) for n>8;
%C A110555 T(n,9) = -A000582(n-1) for n>9;
%C A110555 T(n,10) = A001287(n-1) for n>10;
%C A110555 T(n,11) = -A001288(n-1) for n>11;
%C A110555 T(n,12) = A010965(n-1) for n>12;
%C A110555 T(n,13) = -A010966(n-1) for n>13;
%C A110555 T(n,14) = A010967(n-1) for n>14;
%C A110555 T(n,15) = -A010968(n-1) for n>15;
%C A110555 T(n,16) = A010969(n-1) for n>16.
%C A110555 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 
               0, ...] DELTA [0, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator 
               defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 05 2005
%H A110555 <a href="Sindx_Pas.html#Pascal">Index entries for triangles and arrays 
               related to Pascal's triangle</a>
%F A110555 T(n, k) = binomial(n-1, k)*(-1)^k, 0<=k<n, T(n, n)=0^n.
%Y A110555 Cf. A008949, A007318.
%Y A110555 Sequence in context: A082601 A077593 A119337 this_sequence A071919 A097805 
               A167763
%Y A110555 Adjacent sequences: A110552 A110553 A110554 this_sequence A110556 A110557 
               A110558
%K A110555 sign,tabl
%O A110555 1,8
%A A110555 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005