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.

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

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