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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 (list; table; graph; refs; listen; history; text; internal format)
 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 A180165 Adjacent sequences: A208594 A208595 A208596 * A208598 A208599 A208600 KEYWORD nonn,tabl AUTHOR R. H. Hardin, Feb 29 2012 STATUS approved

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Last modified March 28 23:19 EDT 2023. Contains 361596 sequences. (Running on oeis4.)