A143158 Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j).

1, 0, -1, -1, -2, -1, -1, -2, -1, 0, -2, -3, -2, -1, -1, -1, -2, -1, 0, 0, 1, -2, -3, -2, -1, -1, 0, -1, -2, -3, -2, -1, -1, 0, -1, 0, -2, -3, -2, -1, -1, 0, -1, 0, 0, -1, -2, -1, 0, 0, 1, 0, 1, 1, 1, -2, -3, -2, -1, -1, 0, -1, 0, 0, 0, -1, -2, -3, -2, -1, -1, 0, -1, 0, 0, 0, -1, 0, -3, -4, -3, -2, -2, -1, -2, -1, -1, -1, -2, -1, -1, -2, -3, -2, -1

Offset: 1

Author

Gary W. Adamson, Jul 27 2008

Keywords

• tabl
• sign
• look

Comments

Right border gives A008683.

Left border gives A002321.

Row sums give A068340.

Examples

			First few rows of the triangle =
   1;
   0, -1;
  -1, -2, -1;
  -1, -2, -1,  0;
  -2, -3, -2, -1, -1;
  -2, -3, -2, -1, -1, 0, -1;
  -2, -3, -2, -1, -1, 0, -1, 0;
  -1, -2, -1,  0,  0, 1,  0, 1, 1, 1;
  ...
For example, T(5,3) = (-2) = Sum(-1, 0, -1), since mu(n) = 1, -1, -1, 0, -1, ...
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Crossrefs

Cf. A008683, A002321, A068340, A054527.

Programs

Haskell

import Data.List (tails)
a143158 n k = a143158_tabl !! (n-1) !! (k-1)
a143158_row n = a143158_tabl !! (n-1)
a143158_tabl = map (map sum . init . tails) a054527_tabl
-- Reinhard Zumkeller, Sep 04 2015

Mathematica

Table[Sum[MoebiusMu@ j, {j, k, n}], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Dec 17 2015 *)

PARI

T(n,k) = sum(j=k,n,moebius(j))

Formula

Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j), where mu(n) = A008683.

T(n, k) = A000012(n) * (A008683(n) * 0^(n-k)) * A000012(n).

Extensions

47th term = T(10,2) corrected by Reinhard Zumkeller, Sep 04 2015