%I #18 Jul 28 2019 21:03:24
%S 1,1,1,1,-1,1,1,1,1,1,1,1,-2,-1,1,1,1,1,1,1,1,1,1,1,-3,1,-1,1,1,1,1,1,
%T 1,-2,1,1,1,1,1,1,-4,1,1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-5,1,-3,
%U -2,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-6,1,1,1,1,-1,1,1,1,1,1,1,1,1,1
%N Square array T(n,k) read by antidiagonals up: T(n,k) = 1 if n=1; otherwise if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.
%C Replace the first column in A077049 with any k-th column in A177121 (this array) to get a new array. Then the matrix inverse of the new array will have the k-th column of A054535 (Ramanujan sum) as its first column.
%C Obtained from A176079 by transposing, flipping signs, and adding a lower triangle of all -1's. - _R. J. Mathar_, Jul 08 2011
%H Antti Karttunen, <a href="/A177121/b177121.txt">Table of n, a(n) for n = 1..65703 (the first 362 antidiagonals of array)</a>
%F T(n,k) = 1 if n=1; otherwise, if n divides k then T(n,k) = -n+1; otherwise T(n,k) = 1.
%e Table begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...
%e 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, ...
%e 1, 1, 1, -3, 1, 1, 1, -3, 1, 1, ...
%e 1, 1, 1, 1, -4, 1, 1, 1, 1, -4, ...
%e 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, -7, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, -8, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, -9, ...
%e ...
%o (Excel) =if(row()=1;1;if(mod(column();row())=0;-row()+1;1))
%o (PARI)
%o up_to = 65703; \\ = binomial(362+1,2)
%o A177121sq(row,col) = if(1==row,1,if(!(col%row),(1-row),1));
%o A177121list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, if(i++ > up_to, return(v)); v[i] = A177121sq((a-(col-1)),col))); (v); };
%o v177121 = A177121list(up_to);
%o A177121(n) = v177121[n]; \\ _Antti Karttunen_, Sep 25 2018
%Y Cf. A051731, A054535, A077049, A176079.
%K sign,tabl,look
%O 1,13
%A _Mats Granvik_ and _Gary W. Adamson_, May 03 2010