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A154990 Triangle read by rows. Main diagonal is positive. The rest of the terms are negative.

%I A154990 #19 Mar 07 2021 03:05:44
%S A154990 1,-1,1,-1,-1,1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,
%T A154990 -1,-1,1,-1,-1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,
%U A154990 -1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,-1,-1
%N A154990 Triangle read by rows. Main diagonal is positive. The rest of the terms are negative.
%C A154990 Triangle can be used in matrix inverses. Signs in columns as in A153881.
%C A154990 Iff n is a triangular number, a(n)=1; otherwise, a(n)=-1. (This is explicitly implemented in the second Mathematica program below.) - _Harvey P. Dale_, Apr 27 2014
%H A154990 G. C. Greubel, <a href="/A154990/b154990.txt">Rows n = 1..30 of the triangle, flattened</a>
%F A154990 From _G. C. Greubel_, Mar 06 2021: (Start)
%F A154990 T(n, k) = -1 with T(n, n) = 1.
%F A154990 Sum_{k=1..n} T(n, k) = 2-n = -A023443(n-1) = -A023444(n). (End)
%e A154990 Table begins:
%e A154990    1;
%e A154990   -1,  1;
%e A154990   -1, -1,  1;
%e A154990   -1, -1, -1,  1;
%e A154990   -1, -1, -1, -1,  1;
%e A154990   -1, -1, -1, -1, -1,  1;
%e A154990   -1, -1, -1, -1, -1, -1, 1;
%p A154990 A154990 := proc(n,k)
%p A154990     option remember;
%p A154990     if k = n then
%p A154990         1;
%p A154990     elif k > n then
%p A154990         0;
%p A154990     else
%p A154990         -1 ;
%p A154990     end if;
%p A154990 end proc:
%p A154990 seq(seq(A154990(n,k),k=1..n),n=1..12) ; # _R. J. Mathar_, Sep 16 2017
%t A154990 Flatten[Table[PadLeft[{1},n,-1],{n,15}]] (* or *) With[{tr=Accumulate[ Range[ 15]]}, Table[If[MemberQ[tr,n],1,-1],{n,Last[tr]}]] (* _Harvey P. Dale_, Apr 27 2014 *)
%o A154990 (Sage) flatten([[1 if k==n else -1 for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 06 2021
%o A154990 (Magma) [k eq n select 1 else -1: k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 06 2021
%Y A154990 Cf. A023443, A023444.
%K A154990 sign,easy,tabl
%O A154990 1,1
%A A154990 _Mats Granvik_, Jan 18 2009