0,18

T(h,k) is the number of classes of aperiodic binary words of k ones and h-k zeros; words u,v are in the same class if v is a cyclic permutation of u (e.g., u=111000, v=110001) and a word is aperiodic if not a juxtaposition of 2 or more identical subwords.

T(2n, n), T(2n+1, n), T(n, 3) match A022553, A000108, A001840, respectively. Row sums match A001037.

From R. J. Mathar, Jul 31 2008: (Start)

This triangle may also be regarded as the square array A(r,n), the n-th term of the r-th Witt transform of the all-1 sequence, r >= 1, n >= 0, read by antidiagonals:

This array begins as follows:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63

0 1 2 5 8 14 20 30 40 55 70 91 112 140 168 204 240 285 330

0 1 3 7 14 25 42 66 99 143 200 273 364 476 612 775 969 1197 1463

0 1 3 9 20 42 75 132 212 333 497 728 1026 1428 1932 2583 3384 4389 5598

0 1 4 12 30 66 132 245 429 715 1144 1768 2652 3876 5537 7752 10659 14421 19228

0 1 4 15 40 99 212 429 800 1430 2424 3978 6288 9690 14520 21318 30624 43263 60060

0 1 5 18 55 143 333 715 1430 2700 4862 8398 13995 22610 35530 54477 81719 120175

0 1 5 22 70 200 497 1144 2424 4862 9225 16796 29372 49742 81686 130750 204248

0 1 6 26 91 273 728 1768 3978 8398 16796 32065 58786 104006 178296 297160 482885

0 1 6 30 112 364 1026 2652 6288 13995 29372 58786 112632 208012 371384 643842

0 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400023 742900 1337220

0 1 7 40 168 612 1932 5537 14520 35530 81686 178296 371384 742900 1432613 2674440

...

It is essentially symmetric: A(r,r+i) = A(r,r-i+1).

Some of the diagonals are:

A(r,r+1): A000108

A(r,r): A022553

A(r,r-1): A000108

A(r,r+2): A000150

A(r,r+3): A050181

A(r,r+4): A050182

A(r,r+5): A050183

A(r,r-2): A000150 (End)

Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} A054533(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = T(n + k, k). - Petros Hadjicostas, Jul 09 2019

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Freddy Cachazo and Nick Early, Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes, arXiv:2010.09708 [math.CO], 2020.

A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238; MR1691428 (2000c:13006).

A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188; MR1719140 (2000j:05009). See p. 174. - N. J. A. Sloane, Aug 06 2014

M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.

Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.

Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

P. Stanica and S. Maitra, Rotation symmetric boolean functions - count and cryptographic properties, Disc. Appl. Math. 156 (2008) 1567-1580; see h_{n,w} in eq. (3).

T(h, k) = 1 for (h, k) in {(0, 0), (1, 0), (1, 1)}; T(h, k) = 0 if h >= 2 and k = 0 or k = h. Otherwise, T(h, k) = (1/h)*(C(h, k)-S(h, k)), where S(h, k) = Sum_{d <= 2, d|h, d|k} (h/d)*T(h/d, k/d).

1 - x - y = Product_{i,j} (1 - x^i * y^j)^T(i+j, j) where i >= 0, j >= 0 are not both zero. - Michael Somos, Jul 03 2004

The prime rows are given by (1+x)^p/p with noninteger coefficients rounded to zero. E.g., for h = 2 below, (1 + x)^2/2 = (1 + 2*x + x^2)/2 = 0.5 + x + 0.5*x^2 gives (0,1,0). - Tom Copeland, Oct 21 2014

T(n,k) = (1/n) * Sum_{d | gcd(n,k)} mu(d) * binomial(n/d, k/d), for n > 0. - Andrew Howroyd, Mar 26 2017

From Petros Hadjicostas, Jun 16 2019: (Start)

O.g.f. for column k >= 1: (x^k/k) * Sum_{d|k} mu(d)/(1 - x^d)^(k/d).

Bivariate o.g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = 1 - Sum_{d >= 1} (mu(d)/d) *log(1 - x^d * (1 + y^d)).

(End)

Triangle begins with:

h=0: 1

h=1: 1, 1

h=2: 0, 1, 0

h=3: 0, 1, 1, 0

h=4: 0, 1, 1, 1, 0

h=5: 0, 1, 2, 2, 1, 0

h=6: 0, 1, 2, 3, 2, 1, 0

h=7: 0, 1, 3, 5, 5, 3, 1, 0

h=8: 0, 1, 3, 7, 8, 7, 3, 1, 0

h=9: 0, 1, 4, 9, 14, 14, 9, 4, 1, 0

...

T(6,3) counts classes {111000},{110100},{110010}, each of 6 aperiodic. The class {100100} contains 3 periodic words, counted by T(3,1) as {100}, consisting of 3 aperiodic words 100,010,001.

A := proc(r, n) local gf, d, genf; genf := 1/(1-x) ; gf := 0 ; for d in numtheory[divisors](r) do gf := gf + numtheory[mobius](d)*(subs(x= x^d, genf))^(r/d) ; od: gf := expand(gf/r) ; coeftayl(gf, x=0, n) ; end proc:

A051168 := proc(n, k) if n<=1 then 1; elif n=0 or n=k then 0; else A(n-k, k) ; end if;

end proc:

seq(seq(A051168(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Mar 29 2011

Table[If[n===0, 1, 1/n Plus@@(MoebiusMu[ # ]Binomial[n/#, k/# ]&/@ Divisors[GCD[n, k]])], {n, 0, 12}, {k, 0, n}] (* Wouter Meeussen, Jul 20 2008 *)

(PARI) {T(n, k) = local(A, ps, c); if( k<0 || k>n, 0, if( n==0 && k==0, 1, A = x * O(x^n) + y * O(y^n); ps = 1 - x - y + A; for( m=1, n, for( i=0, m, c = polcoeff( polcoeff(ps, i, x), m-i, y); if( m==n && i==k, break(2), ps *= (1 -y^(m-i) * x^i + A)^c))); -c))} /* Michael Somos, Jul 03 2004 */

(PARI) T(n, k) = if (n==0, 1, (1/n) * sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d)));

tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 16 2018

Columns 1-11: A000012, A004526(n-1), A001840(n-4), A006918(n-4), A011795(n-1), A011796(n-6), A011797(n-1), A031164(n-9), A011845, A032168, A032169. See also A000150.

Cf. A047996, A052307, A052314, A092964, A185158, A123223, A124814.

Sequence in context: A234044 A323258 A219489 * A367895 A368127 A368122

Adjacent sequences: A051165 A051166 A051167 * A051169 A051170 A051171

nonn,tabl

approved