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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; 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

Table of n, a(n) for n=0..90.

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)])

CROSSREFS

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

AUTHOR

Peter Luschny, Sep 10 2018

STATUS

approved

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Last modified September 22 02:38 EDT 2021. Contains 347605 sequences. (Running on oeis4.)