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%I #9 Dec 17 2019 05:34:14
%S 1,0,1,0,-1,1,0,1,-1,1,0,-1,0,-1,1,0,1,1,0,-1,1,0,-1,-1,0,0,-1,1,0,1,
%T 0,1,0,0,-1,1,0,-1,1,-1,0,0,0,-1,1,0,1,-1,0,1,0,0,0,-1,1,0,-1,0,0,-1,
%U 0,0,0,0,-1,1,0,1,1,1,0,1,0,0,0,0,-1,1,0,-1,-1,-1,0,-1,0,0,0,0,0,-1,1,0,1
%N Triangle T(n,k) read by rows: A051731(n,k) - A051731(n-1,k).
%C The recurrence for this triangle is similar to the recurrence in A177517. Cumulative column sums give table A051731.
%F T(n,1)=A000007, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, -1, 1;
%e 0, 1, -1, 1;
%e 0, -1, 0, -1, 1;
%e 0, 1, 1, 0, -1, 1;
%e 0, -1, -1, 0, 0, -1, 1;
%e 0, 1, 0, 1, 0, 0, -1, 1;
%e 0, -1, 1, -1, 0, 0, 0, -1, 1;
%e 0, 1, -1, 0, 1, 0, 0, 0, -1, 1;
%e 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 1;
%e 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, -1, 1;
%e 0, -1, -1, -1, 0, -1, 0, 0, 0, 0, 0, -1, 1;
%e 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1;
%o (Excel cell formula, American) =if(column()=1, if(row()>1,0,1), if(row()>=column(), sum(indirect(address(row()-column()+1, column()-1, 4)&":"&address(row()-1, column()-1, 4), 4))-sum(indirect(address(row()-column()+1, column(), 4)&":"&address(row()-1, column(), 4), 4)), 0))
%o (Excel cell formula, European)=if(column()=1; if(row()>1;0;1); if(row()>=column(); sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4)); 0))
%Y Matrix inverse of A134540. Cf. A177517.
%K sign,tabl
%O 1,1
%A _Mats Granvik_, May 16 2010
%E Typo in sequence (erroneous comma) corrected by _N. J. A. Sloane_, May 22 2010
%E Edited by _Mats Granvik_, Dec 11 2010