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A208597
T(n,k) = number of n-bead necklaces labeled with numbers -k..k not allowing reversal, with sum zero.
14
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
Rows 3-7 are A002061(n+1), A208598, A208599, A208600, A208601.
Main diagonal is A208590.
Sequence in context: A352001 A208337 A208335 * A179943 A089944 A374738
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Feb 29 2012
STATUS
approved