OFFSET
1,2
COMMENTS
Column k is Dirichlet convolution of mu(n) with k^n.
The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k^2+k^4.
LINKS
Alois P. Heinz, Antidiagonals n = 1..141, flattened
C. J. Smyth, A coloring proof of a generalisation of Fermat's little theorem, Amer. Math. Monthly 93, No. 6 (1986), pp. 469-471.
FORMULA
T(n,k) = Sum_{d|n} k^d * mu(n/d).
T(n,k) = k^n - Sum_{d<n,d|n} T(d,k).
T(n,k) = A143325(n,k) * k.
T(n,k) = A074650(n,k) * n.
So Sum_{d|n} k^d * mu(n/d) == 0 (mod n), this is a generalization of Fermat's little theorem k^p - k == 0 (mod p) for primes p to an arbitrary modulus n (see the Smyth link). - Franz Vrabec, Feb 09 2021
EXAMPLE
T(2,3)=6, because there are 6 primitive words of length 2 over 3-letter alphabet {a,b,c}: ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
1, 2, 3, 4, 5, ...
0, 2, 6, 12, 20, ...
0, 6, 24, 60, 120, ...
0, 12, 72, 240, 600, ...
0, 30, 240, 1020, 3120, ...
MAPLE
with(numtheory): f0:= proc(n) option remember; unapply(k^n-add(f0(d)(k), d=divisors(n)minus{n}), k) end; T:= (n, k)-> f0(n)(k); seq(seq(T(n, 1+d-n), n=1..d), d=1..12);
MATHEMATICA
f0[n_] := f0[n] = Function [k, k^n - Sum[f0[d][k], {d, Complement[Divisors[n], {n}]}]]; t[n_, k_] := f0[n][k]; Table[Table[t[n, 1 + d - n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 07 2008
STATUS
approved