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

1,1

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

Links

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

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)

Example

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;

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 *)

Prog

(Sage) flatten([[1 if k==n else -1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021
(Magma) [k eq n select 1 else -1: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

Crossrefs

Cf. A023443, A023444.

Sequence in context: A057077 A262725 A070748 * A209615 A242179 A319117

Adjacent sequences: A154987 A154988 A154989 * A154991 A154992 A154993

Keyword

sign,easy,tabl

Author

Mats Granvik, Jan 18 2009

Status

approved