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A110540
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Invertible triangle: T(n,k) = number of k-ary Lyndon words of length n-k+1 with trace 1 modulo k.
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5
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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, 0, 93, 1960, 7280, 9750, 6665, 2792, 819, 180, 33, 5, 1
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OFFSET
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1,12
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COMMENTS
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An invertible number triangle related to Lyndon words of trace 1.
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LINKS
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FORMULA
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T(n, k) = Sum_{d | n-k+1, gcd(d, k)=1} mu(d)*k^((n-k+1)/d))/(k*(n-k+1)).
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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