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
Examples
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
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 185 terms from R. H. Hardin)
Crossrefs
Programs
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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 *) -
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
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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
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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
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.