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A143158
Triangle read by rows, T(n,k) = Sum_{j=k..n} mu(j).
2
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
1,5
COMMENTS
Right border gives A008683.
Left border gives A002321.
Row sums give A068340.
LINKS
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).
EXAMPLE
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, ...
MATHEMATICA
Table[Sum[MoebiusMu@ j, {j, k, n}], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Dec 17 2015 *)
PROG
(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
(PARI) T(n, k) = sum(j=k, n, moebius(j))
CROSSREFS
KEYWORD
tabl,sign,look
AUTHOR
Gary W. Adamson, Jul 27 2008
EXTENSIONS
47th term = T(10,2) corrected by Reinhard Zumkeller, Sep 04 2015
STATUS
approved