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

%I A143158 #20 Jan 14 2023 09:29:32
%S A143158 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 A143158 -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 A143158 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 A143158 Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j).
%C A143158 Right border gives A008683.
%C A143158 Left border gives A002321.
%C A143158 Row sums give A068340.
%H A143158 Reinhard Zumkeller, <a href="/A143158/b143158.txt">Rows n = 1..125 of triangle, flattened</a>
%F A143158 Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j), where mu(n) = A008683.
%F A143158 T(n, k) = A000012(n) * (A008683(n) * 0^(n-k)) * A000012(n).
%e A143158 First few rows of the triangle =
%e A143158    1;
%e A143158    0, -1;
%e A143158   -1, -2, -1;
%e A143158   -1, -2, -1,  0;
%e A143158   -2, -3, -2, -1, -1;
%e A143158   -2, -3, -2, -1, -1, 0, -1;
%e A143158   -2, -3, -2, -1, -1, 0, -1, 0;
%e A143158   -1, -2, -1,  0,  0, 1,  0, 1, 1, 1;
%e A143158   ...
%e A143158 For example, T(5,3) = (-2) = Sum(-1, 0, -1), since mu(n) = 1, -1, -1, 0, -1, ...
%t A143158 Table[Sum[MoebiusMu@ j, {j, k, n}], {n, 14}, {k, n}] // Flatten (* _Michael De Vlieger_, Dec 17 2015 *)
%o A143158 (Haskell)
%o A143158 import Data.List (tails)
%o A143158 a143158 n k = a143158_tabl !! (n-1) !! (k-1)
%o A143158 a143158_row n = a143158_tabl !! (n-1)
%o A143158 a143158_tabl = map (map sum . init . tails) a054527_tabl
%o A143158 -- _Reinhard Zumkeller_, Sep 04 2015
%o A143158 (PARI) T(n,k) = sum(j=k,n,moebius(j))
%Y A143158 Cf. A008683, A002321, A068340, A054527.
%K A143158 tabl,sign,look
%O A143158 1,5
%A A143158 _Gary W. Adamson_, Jul 27 2008
%E A143158 47th term = T(10,2) corrected by _Reinhard Zumkeller_, Sep 04 2015