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A103447
Triangle read by rows: T(n,k) = Moebius(binomial(n,k)) (0 <= k <= n).
5
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).
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
KEYWORD
sign,tabl
AUTHOR
Emeric Deutsch, Feb 06 2005
STATUS
approved