A103447 - OEIS

%I #20 Sep 08 2022 08:45:16

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

%C T(2*n, n) = 0 for all n except n=0, 1, 2 and 4 (Granville and Ramare).

%H G. C. Greubel, <a href="/A103447/b103447.txt">Rows n = 0..100 of the triangle, flattened</a>

%H 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 T(n, k) = Moebius(binomial(n, k)) (0 <= k <= n).

%F T(n, k) = A008683(A007318(n, k)).

%F Sum_{k=0..n} T(n, k) = A103448(n).

%e T(3,2)=-1 because binomial(3,2)=3 and Moebius(3)=-1.

%e Triangle begins:

%e   1;

%e   1,  1;

%e   1, -1,  1;

%e   1, -1, -1,  1;

%e   1,  0,  1,  0,  1;

%e   1, -1,  1,  1, -1,  1;

%p 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 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 (PARI) T(n,k) = moebius(binomial(n,k))

%o for(n=0, 15, for(k=0, n, print1(T(n,k)", "))) \\ _Charles R Greathouse IV_, Nov 03 2014

%o (Magma) [MoebiusMu(Binomial(n, k)): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 16 2021

%o (Sage)

%o def T(n, k): return moebius(binomial(n, k))

%o flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jun 16 2021

%Y Cf. A007318, A008683, A103448 (row sums), A103449.

%K sign,tabl

%O 0,1

%A _Emeric Deutsch_, Feb 06 2005