A103448 -id:A103448 - cp's OEIS frontend

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

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, 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, 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

Offset: 0

Emeric Deutsch, Feb 06 2005

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

			T(3,2)=-1 because binomial(3,2)=3 and Moebius(3)=-1.
Triangle begins:
  1;
  1,  1;
  1, -1,  1;
  1, -1, -1,  1;
  1,  0,  1,  0,  1;
  1, -1,  1,  1, -1,  1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI].

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

[MoebiusMu(Binomial(n, k)): k in [0..n], n in [0..15]]; // G. C. Greubel, Jun 16 2021

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[n_, k_]:= MoebiusMu[Binomial[n, k]]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 16 2021 *)

T(n,k) = moebius(binomial(n,k))
for(n=0, 15, for(k=0, n, print1(T(n,k)", "))) \\ Charles R Greathouse IV, Nov 03 2014

def T(n, k): return moebius(binomial(n, k))
flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jun 16 2021

T(n, k) = Moebius(binomial(n, k)) (0 <= k <= n).
T(n, k) = A008683(A007318(n, k)).
Sum_{k=0..n} T(n, k) = A103448(n).

A103449 Values of k such that Sum_{m=0..k} Moebius(binomial(k,m)) = 0.

Original entry on oeis.org

3, 12, 24, 29, 34, 40, 54, 60, 67, 68, 75, 86, 93, 97, 102, 119, 125, 131, 133, 142, 152, 157, 160, 163, 164, 168, 170, 172, 189, 193, 197, 208, 210, 220, 221, 228, 229, 246, 251, 255, 257, 261, 270, 275, 280, 293, 296, 307, 308, 313, 315, 332, 337, 338, 340

Offset: 1

Emeric Deutsch, Feb 07 2005

nonn

Values of k such that A103448(k) = 0.

			12 belongs to the sequence because the only squarefree values of binomial(12,m) are 1, 2*3*11, 2*3*11, 1, on which the Mobius function takes the values 1,-1,-1,1, respectively.
8 does not belong to the sequence because the only squarefree value of binomial(8,m) are 1, 2*5*7, 1, on which the Moebius function takes the values 1,-1,1, respectively.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A008683, A103447, A103448.

A103448[n_]:= A103448[n]= Sum[MoebiusMu[Binomial[n, k]], {k, 0, n}];
f:= Table[A103448[n], {n, 0, 1050}];
Select[Range[0, 1000], f[[#]] == 0 &] - 1  (* G. C. Greubel, Jun 17 2021 *)

is(k) = sum(m=0, k, moebius(binomial(k, m)))==0 \\ Felix Fröhlich, Jun 18 2021

cp's OEIS Frontend

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

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