OFFSET
1,12
COMMENTS
An invertible number triangle related to Lyndon words of trace 1.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
Frank Ruskey, Number of q-ary Lyndon words with given trace mod q
Frank Ruskey, Number of Lyndon words over GF(q) with given trace
FORMULA
T(n, k) = (Sum_{d | n-k+1, gcd(d, k)=1} mu(d)*k^((n-k+1)/d))/(k*(n-k+1)).
EXAMPLE
Rows begin
1;
0, 1;
0, 1, 1;
0, 1, 1, 1;
0, 2, 3, 2, 1;
0, 3, 6, 5, 2, 1;
0, 5, 16, 16, 8, 3, 1;
0, 9, 39, 51, 30, 12, 3, 1;
0, 16, 104, 170, 125, 54, 16, 4, 1;
0, 28, 270, 585, 516, 259, 84, 21, 4, 1;
0, 51, 729, 2048, 2232, 1296, 480, 128, 27, 5, 1;
MATHEMATICA
T[n_, k_]:=Sum[Boole[GCD[d, k] == 1] MoebiusMu[d] k^((n - k + 1)/d), {d, Divisors[n - k + 1]}] /(k(n - k + 1)); Flatten[Table[T[n, k], {n, 12}, {k, n}]] (* Indranil Ghosh, Mar 27 2017 *)
PROG
(PARI)
for(n=1, 11, for(k=1, n, print1( sum(d=1, n-k+1, if(Mod(n-k+1, d)==0 && gcd(d, k)==1, moebius(d)*k^((n-k+1)/d), 0)/(k*(n-k+1)) ), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jul 25 2005
EXTENSIONS
Name clarified by Andrew Howroyd, Mar 26 2017
STATUS
approved