A127899 Transform related to the harmonic series.

1, -2, 2, 0, -3, 3, 0, 0, -4, 4, 0, 0, 0, -5, 5, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, 0, -7, 7, 0, 0, 0, 0, 0, 0, -8, 8, 0, 0, 0, 0, 0, 0, 0, -9, 9, 0, 0, 0, 0, 0, 0, 0, 0, -10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 12

Offset: 1

Author

Gary W. Adamson, Feb 04 2007

Keywords

• tabl
• sign
• easy

Comments

This transform is the inverse of a triangle in which each row has n terms of the harmonic series; i.e., the inverse of: 1; 1, 1/2; 1, 1/2, 1/3; ...

Eigensequence of the unsigned triangle = A002467 starting (1, 4, 15, 76, 455, ...). - Gary W. Adamson, Dec 29 2008

Table T(n,k) read by antidiagonals. T(1,1)=1, T(n,1) = n (for n>1), T(n,2) = -n, T(n,k) = 0, k > 2. - Boris Putievskiy, Jan 17 2013

Examples

			First few rows of the triangle are:
1;
-2, 2;
0, -3, 3;
0, 0, -4, 4;
0, 0, 0, -5, 5;
0, 0, 0, 0, -6, 6;
0, 0, 0, 0, 0, -7, 7;
...
From _Boris Putievskiy_, Jan 17 2013: (Start)
The start of the sequence as table:
1..-1..0..0..0..0..0...
1..-2..0..0..0..0..0...
2..-3..0..0..0..0..0...
3..-4..0..0..0..0..0...
4..-5..0..0..0..0..0...
5..-6..0..0..0..0..0...
6..-7..0..0..0..0..0...
...
The start of the sequence as triangle array read by rows:
1;
-1,1;
0,-2,2;
0,0,-3,3;
0,0,0,-4,4;
0,0,0,0,-5,5;
0,0,0,0,0,-6,6;
0,0,0,0,0,0,-7,7;
...
Row number r (r>4) contains (r-2) times '0', then '-r' and 'r'. (End)
		

Links

• Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A002467.

Programs

Haskell

a127899 n k = a127899_tabl !! (n-1) !! (k-1)
a127899_row n = a127899_tabl !! (n-1)
a127899_tabl = map reverse ([1] : xss) where
   xss = iterate (\(u : v : ws) -> u + 1 : v - 1 : ws ++ [0]) [2, -2]
-- Reinhard Zumkeller, Nov 14 2014

Maple

A127899 := proc(n,k)
    if k = n then
        n;
    elif k = n-1 then
        -n;
    else
        0;
    end if;
end proc:
seq(seq( A127899(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Jul 19 2024

Mathematica

Table[Module[{t = Floor[(-1 + Sqrt[8 n - 7])/2], i}, i = n - t (t + 1)/2; Floor[(i + 2)/(t + 2)] (t + 1) (-1)^(i + t + 1)], {n, 78}] (* or *)
Table[If[n == 1, {n}, ConstantArray[0, n - 2]~Join~{-n, n}], {n, 12}] // Flatten (* Michael De Vlieger, Feb 11 2017 *)

Python

from math import isqrt
def A127899(n): return -(isqrt(n<<3)+1>>1)**2+(m:=isqrt(n+1<<3)+1>>1)*((m<<1)-1)+(k:=isqrt(n+2<<3)+1>>1)*(1-k)>>1 # Chai Wah Wu, Jun 08 2025

Formula

Triangle, a(1) = 1; by rows, (n-2) zeros followed by -n, n.

From Boris Putievskiy, Jan 17 2013: (Start)

a(n) = floor((A002260(n)+2)/(A003056(n)+2))*(A003056(n)+1)*(-1)^(A002260(n)+A003056(n)+1), n>0.

a(n) = floor((i+2)/(t+2))*(t+1)*(-1)^(i+t+1), where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)

a(n) = floor(-1/2*A002024(n)^2 + A002024(n+1)^2-1/2*A002024(n+1) + 1/2*A002024(n+2) - 1/2*A002024(n+2)^2). - Brian Tenneson, Feb 10 2017