A208597 T(n,k) = number of n-bead necklaces labeled with numbers -k..k not allowing reversal, with sum zero.

1, 1, 2, 1, 3, 3, 1, 4, 7, 6, 1, 5, 13, 23, 11, 1, 6, 21, 60, 77, 26, 1, 7, 31, 125, 291, 297, 57, 1, 8, 43, 226, 791, 1564, 1163, 142, 1, 9, 57, 371, 1761, 5457, 8671, 4783, 351, 1, 10, 73, 568, 3431, 14838, 39019, 49852, 20041, 902, 1, 11, 91, 825, 6077, 34153, 129823

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 185 terms from R. H. Hardin)

Formula

T(n,k) = Sum_{d|n} phi(n/d) * A201552(d, k). - Andrew Howroyd, Oct 14 2017

Empirical for row n:

n=1: a(k) = 1.

n=2: a(k) = k + 1.

n=3: a(k) = k^2 + k + 1.

n=4: a(k) = (4/3)*k^3 + 2*k^2 + (5/3)*k + 1.

n=5: a(k) = (23/12)*k^4 + (23/6)*k^3 + (37/12)*k^2 + (7/6)*k + 1.

n=6: a(k) = (44/15)*k^5 + (22/3)*k^4 + (23/3)*k^3 + (14/3)*k^2 + (12/5)*k + 1.

n=7: a(k) = (841/180)*k^6 + (841/60)*k^5 + (325/18)*k^4 + (51/4)*k^3 + (949/180)*k^2 + (37/30)*k + 1.

Example

Table starts
...1....1.....1......1.......1.......1........1........1........1.........1
...2....3.....4......5.......6.......7........8........9.......10........11
...3....7....13.....21......31......43.......57.......73.......91.......111
...6...23....60....125.....226.....371......568......825.....1150......1551
..11...77...291....791....1761....3431.....6077....10021....15631.....23321
..26..297..1564...5457...14838...34153....69784...130401...227314....374825
..57.1163..8671..39019..129823..353333...833253..1764925..3438877...6267735
.142.4783.49852.288317.1172298.3770475.10259448.24627705.53630854.108036775

Mathematica

comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r-1-i*m, k-1]*Binomial[k, i], {i, 0, Floor[(r-k)/m]}]; a[n_, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2k+1, #]&]/n; Table[a[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd's PARI code *)

Prog

(PARI)
comps(r, m, k)=sum(i=0, floor((r-k)/m), (-1)^i*binomial(r-1-i*m, k-1)*binomial(k, i));
a(n, k)=sumdiv(n, d, eulerphi(n/d)*comps(d*(k+1), 2*k+1, d))/n;
for(n=1, 8, for(k=1, 10, print1(a(n, k), ", ")); print()); \\ Andrew Howroyd, May 16 2017
(Python)
from sympy import binomial, divisors, totient, floor
def comps(r, m, k): return sum([(-1)**i*binomial(r - 1 - i*m, k - 1)*binomial(k, i) for i in range(floor((r - k)/m) + 1)])
def a(n, k): return sum([totient(n//d)*comps(d*(k + 1), 2*k + 1, d) for d in divisors(n)])//n
for n in range(1, 12): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Nov 07 2017, after PARI code
(R)
require(numbers)
comps <- function(r, m, k) {
  S <- numeric()
  for (i in 0:floor((r-k)/m)) S <- c(S, (-1)^i*choose(r-1-i*m, k-1)*choose(k, i))
  return(sum(S))
}
a <- function(n, k) {
  S <- numeric()
  for (d in divisors(n)) S <- c(S, eulersPhi(n/d)*comps(d*(k+1), 2*k+1, d))
  return(sum(S)/n)
}
for (n in 1:11) {
  for (k in 1:n) {
    print(a(k, n-k+1))
  }
} # Indranil Ghosh, Nov 07 2017, after PARI code

Crossrefs

Columns 1-7 are A208602, A208591, A208592, A208593, A208594, A208595, A208596.

Rows 3-7 are A002061(n+1), A208598, A208599, A208600, A208601.

Main diagonal is A208590.

Cf. A201552, A286928, A208825.

Sequence in context: A352001 A208337 A208335 * A179943 A089944 A374738

Adjacent sequences: A208594 A208595 A208596 * A208598 A208599 A208600

Keyword

nonn,tabl

Author

R. H. Hardin, Feb 29 2012

Status

approved