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 A143324 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1). 25
 1, 2, 0, 3, 2, 0, 4, 6, 6, 0, 5, 12, 24, 12, 0, 6, 20, 60, 72, 30, 0, 7, 30, 120, 240, 240, 54, 0, 8, 42, 210, 600, 1020, 696, 126, 0, 9, 56, 336, 1260, 3120, 4020, 2184, 240, 0, 10, 72, 504, 2352, 7770, 15480, 16380, 6480, 504, 0, 11, 90, 720, 4032, 16800, 46410, 78120, 65280, 19656, 990, 0 (list; table; graph; refs; listen; history; text; internal format)
 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 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 Columns k=1-10 give: A000007(n-1), A027375, A054718, A054719, A054720, A054721, A218124, A218125, A218126, A218127. Rows n=1-10 give: A000027, A002378(k-1), A007531(k+1), A047928(k+1), A061167, A218130, A133499, A218131, A218132, A218133. Main diagonal gives A252764. Cf. A074650, A143325, A008683, A054525. Sequence in context: A154559 A342239 A269133 * A287416 A097418 A154752 Adjacent sequences:  A143321 A143322 A143323 * A143325 A143326 A143327 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 07 2008 STATUS approved

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Last modified January 18 07:00 EST 2022. Contains 350454 sequences. (Running on oeis4.)