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 A319082 A(n, k) = (1/k)*Sum_{d|k} EulerPhi(d)*n^(k/d) for n >= 0 and k > 0, A(n, 0) = 0, square array read by ascending antidiagonals. 2
 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 3, 1, 0, 0, 4, 6, 4, 1, 0, 0, 5, 10, 11, 6, 1, 0, 0, 6, 15, 24, 24, 8, 1, 0, 0, 7, 21, 45, 70, 51, 14, 1, 0, 0, 8, 28, 76, 165, 208, 130, 20, 1, 0, 0, 9, 36, 119, 336, 629, 700, 315, 36, 1, 0, 0, 10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1, 0, 0, 11, 55, 249, 1044, 3367, 7826, 11165, 8230, 2195, 108, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 REFERENCES D. E. Knuth, Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, Addison-Wesley, 2005. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals) H. Fredricksen and I. J. Kessler, An algorithm for generating necklaces of beads in two colors, Discrete Math. 61 (1986), 181-188. H. Fredricksen and J. Maiorana, Necklaces of beads in k colors and k-ary de Bruijn sequences, Discrete Math. 23(3) (1978), 207-210. Reviewed in MR0523071 (80e:05007). Peter Luschny, Implementation of the FKM algorithm in SageMath and Julia F. Ruskey, C. Savage, and T. M. Y. Wang, Generating necklaces, Journal of Algorithms, 13(3), 1992, 414-430. Index entries for sequences related to necklaces FORMULA A(n, k) = (1/k)*Sum_{i=1..k} n^gcd(i, k) for k > 0. EXAMPLE Array starts: [n\k][0 1 2 3 4 5 6 7 8 9 ...] [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... [1] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... [2] 0, 2, 3, 4, 6, 8, 14, 20, 36, 60, ... [3] 0, 3, 6, 11, 24, 51, 130, 315, 834, 2195, ... [4] 0, 4, 10, 24, 70, 208, 700, 2344, 8230, 29144, ... [5] 0, 5, 15, 45, 165, 629, 2635, 11165, 48915, 217045, ... [6] 0, 6, 21, 76, 336, 1560, 7826, 39996, 210126, 1119796, ... [7] 0, 7, 28, 119, 616, 3367, 19684, 117655, 720916, 4483815, ... MAPLE with(numtheory): A := (n, k) -> `if`(k=0, 0, (1/k)*add(phi(d)*n^(k/d), d=divisors(k))): seq(seq(A(n-k, k), k=0..n), n=0..12); # Alternatively, row-wise printed as a table: T := (n, k) -> `if`(k=0, 0, add(n^igcd(i, k), i=1..k)/k): seq(lprint(seq(T(n, k), k=0..9)), n=0..7); PROG (Sage) def A319082(n, k): return 0 if k == 0 else (1/k)*sum(euler_phi(d)*n^(k//d) for d in divisors(k)) for n in (0..7): print([n], [A319082(n, k) for k in (0..9)]) (PARI) A(n, k)=if(k==0, 0, sumdiv(k, d, eulerphi(d)*n^(k/d))/k) \\ Andrew Howroyd, Jan 05 2024 CROSSREFS Essentially the same table as A075195. A185651(n, k) = n*A(k, n). Main diagonal gives A056665. A054630(n,k) is a subtriangle for n >= 1 and 1 <= k <= n. Sequence in context: A106237 A071675 A221833 * A034365 A103778 A099423 Adjacent sequences: A319079 A319080 A319081 * A319083 A319084 A319085 KEYWORD nonn,tabl AUTHOR Peter Luschny, Sep 10 2018 STATUS approved

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