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A051168
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Triangular array T(h,k) for 0<=k<=h read by rows: T(h,k) = number of binary Lyndon words with k ones and h-k zeros.
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40
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1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 5, 5, 3, 1, 0, 0, 1, 3, 7, 8, 7, 3, 1, 0, 0, 1, 4, 9, 14, 14, 9, 4, 1, 0, 0, 1, 4, 12, 20, 25, 20, 12, 4, 1, 0, 0, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 0, 0, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 0, 0, 1, 6
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OFFSET
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0,18
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COMMENTS
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T(h,k) = 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)
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LINKS
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Table of n, a(n) for n=0..93.
A. Elashvili, 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, 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
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.
Index entries for sequences related to Lyndon words
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FORMULA
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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{(h/d)*T(h/d, k/d): d<=2, d|h, d|k}.
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 non-integer coefficients rounded to zero. E.g., for h=2 below, (1+x)^2/2= (1+2x+x^2)/2=.5 + x +.5 x^2 gves (0,1,0).- Tom Copeland, Oct 21 2014
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EXAMPLE
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Triangle begins with:
h=0: 1
h=1: 1, 1
h=2: 0, 1, 0
h=3: 0, 1, 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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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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.
Sequence in context: A015199 A234044 A219489 * A163528 A239509 A258747
Adjacent sequences: A051165 A051166 A051167 * A051169 A051170 A051171
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling
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STATUS
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approved
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