A143158 - OEIS

%I #20 Jan 14 2023 09:29:32

%S 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,

%T -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,

%U 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

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

%C Right border gives A008683.

%C Left border gives A002321.

%C Row sums give A068340.

%H Reinhard Zumkeller, <a href="/A143158/b143158.txt">Rows n = 1..125 of triangle, flattened</a>

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

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

%e First few rows of the triangle =

%e    1;

%e    0, -1;

%e   -1, -2, -1;

%e   -1, -2, -1,  0;

%e   -2, -3, -2, -1, -1;

%e   -2, -3, -2, -1, -1, 0, -1;

%e   -2, -3, -2, -1, -1, 0, -1, 0;

%e   -1, -2, -1,  0,  0, 1,  0, 1, 1, 1;

%e   ...

%e For example, T(5,3) = (-2) = Sum(-1, 0, -1), since mu(n) = 1, -1, -1, 0, -1, ...

%t Table[Sum[MoebiusMu@ j, {j, k, n}], {n, 14}, {k, n}] // Flatten (* _Michael De Vlieger_, Dec 17 2015 *)

%o (Haskell)

%o import Data.List (tails)

%o a143158 n k = a143158_tabl !! (n-1) !! (k-1)

%o a143158_row n = a143158_tabl !! (n-1)

%o a143158_tabl = map (map sum . init . tails) a054527_tabl

%o -- _Reinhard Zumkeller_, Sep 04 2015

%o (PARI) T(n,k) = sum(j=k,n,moebius(j))

%Y Cf. A008683, A002321, A068340, A054527.

%K tabl,sign,look

%O 1,5

%A _Gary W. Adamson_, Jul 27 2008

%E 47th term = T(10,2) corrected by _Reinhard Zumkeller_, Sep 04 2015