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

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, -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, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Author

Mats Granvik, Jan 18 2009

Keywords

• sign
• easy
• tabl

Comments

Triangle can be used in matrix inverses. Signs in columns as in A153881.

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

Examples

			Table begins:
   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, -1, -1, 1;
		

Links

• G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Crossrefs

Cf. A023443, A023444.

Programs

Magma

[k eq n select 1 else -1: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

Maple

A154990 := proc(n,k)
    option remember;
    if k = n then
        1;
    elif k > n then
        0;
    else
        -1 ;
    end if;
end proc:
seq(seq(A154990(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Sep 16 2017

Mathematica

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 *)

Sage

flatten([[1 if k==n else -1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021

Formula

From G. C. Greubel, Mar 06 2021: (Start)

T(n, k) = -1 with T(n, n) = 1.

Sum_{k=1..n} T(n, k) = 2-n = -A023443(n-1) = -A023444(n). (End)