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

0,1

Comments

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

Links

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].

Formula

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).

Example

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;

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A014295 A344884 A228710 * A354923 A354924 A353799

Adjacent sequences: A103444 A103445 A103446 * A103448 A103449 A103450

Keyword

sign,tabl

Author

Emeric Deutsch, Feb 06 2005

Status

approved