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A103447 Triangle read by rows: T(n,k) = Moebius(binomial(n,k)) (0 <= k <= n).

%I A103447 #20 Sep 08 2022 08:45:16
%S A103447 1,1,1,1,-1,1,1,-1,-1,1,1,0,1,0,1,1,-1,1,1,-1,1,1,1,1,0,1,1,1,1,-1,1,
%T A103447 1,1,1,-1,1,1,0,0,0,-1,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,0,
%U A103447 1,1,1,-1,1,-1,1,1,1,1,-1,1,-1,1,1,0,-1,0,0,0,0,0,0,0,-1,0,1,1,-1,-1,-1,-1,0,0,0,0,-1,-1,-1,-1,1
%N A103447 Triangle read by rows: T(n,k) = Moebius(binomial(n,k)) (0 <= k <= n).
%C A103447 T(2*n, n) = 0 for all n except n=0, 1, 2 and 4 (Granville and Ramare).
%H A103447 G. C. Greubel, <a href="/A103447/b103447.txt">Rows n = 0..100 of the triangle, flattened</a>
%H A103447 A. Granville and O. Ramaré, <a href="http://www.dms.umontreal.ca/~andrew/PDF/ramare.pdf">Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients</a>, Mathematika 43 (1996), 73-107, <a href="http://dx.doi.org/10.1112/S0025579300011608">[DOI]</a>.
%F A103447 T(n, k) = Moebius(binomial(n, k)) (0 <= k <= n).
%F A103447 T(n, k) = A008683(A007318(n, k)).
%F A103447 Sum_{k=0..n} T(n, k) = A103448(n).
%e A103447 T(3,2)=-1 because binomial(3,2)=3 and Moebius(3)=-1.
%e A103447 Triangle begins:
%e A103447   1;
%e A103447   1,  1;
%e A103447   1, -1,  1;
%e A103447   1, -1, -1,  1;
%e A103447   1,  0,  1,  0,  1;
%e A103447   1, -1,  1,  1, -1,  1;
%p A103447 with(numtheory):T:=proc(n,k) if k<=n then mobius(binomial(n,k)) else 0 fi end: for n from 0 to 15 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A103447 T[n_, k_]:= MoebiusMu[Binomial[n, k]]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 16 2021 *)
%o A103447 (PARI) T(n,k) = moebius(binomial(n,k))
%o A103447 for(n=0, 15, for(k=0, n, print1(T(n,k)", "))) \\ _Charles R Greathouse IV_, Nov 03 2014
%o A103447 (Magma) [MoebiusMu(Binomial(n, k)): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 16 2021
%o A103447 (Sage)
%o A103447 def T(n, k): return moebius(binomial(n, k))
%o A103447 flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jun 16 2021
%Y A103447 Cf. A007318, A008683, A103448 (row sums), A103449.
%K A103447 sign,tabl
%O A103447 0,1
%A A103447 _Emeric Deutsch_, Feb 06 2005