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

Author

Peter Luschny, Sep 10 2018

Keywords

• nonn
• tabl

Examples

			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, ...
		

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

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.

Programs

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);

PARI

A(n,k)=if(k==0, 0, sumdiv(k,d, eulerphi(d)*n^(k/d))/k) \\ Andrew Howroyd, Jan 05 2024

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)])

Formula

A(n, k) = (1/k)*Sum_{i=1..k} n^gcd(i, k) for k > 0.