Search: id:A051168
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%I A051168
%S A051168 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,
%T A051168 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,
%U A051168 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
%N A051168 Triangular array T read by rows: T(h,k) = number of classes of aperiodic 
               binary words of k 1's and h-k 0's; 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.
%C A051168 T(h,k)=number of Lyndon words of k 1's and h-k 0's.
%C A051168 T(2n, n), T(2n+1, n), T(n, 3) match A022553, A000108, A001840, respectively. 
               Row sums match A001037.
%C A051168 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
%C A051168 Comment 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:
%C A051168 This array begins as follows:
%C A051168 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%C A051168 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
%C A051168 0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63
%C A051168 0 1 2 5 8 14 20 30 40 55 70 91 112 140 168 204 240 285 330
%C A051168 0 1 3 7 14 25 42 66 99 143 200 273 364 476 612 775 969 1197 1463
%C A051168 0 1 3 9 20 42 75 132 212 333 497 728 1026 1428 1932 2583 3384 4389 5598
%C A051168 0 1 4 12 30 66 132 245 429 715 1144 1768 2652 3876 5537 7752 10659 14421 
               19228
%C A051168 0 1 4 15 40 99 212 429 800 1430 2424 3978 6288 9690 14520 21318 30624 
               43263 60060
%C A051168 0 1 5 18 55 143 333 715 1430 2700 4862 8398 13995 22610 35530 54477 81719 
               120175 173583
%C A051168 0 1 5 22 70 200 497 1144 2424 4862 9225 16796 29372 49742 81686 130750 
               204248 312455 468611
%C A051168 0 1 6 26 91 273 728 1768 3978 8398 16796 32065 58786 104006 178296 297160 
               482885 766935 1193010
%C A051168 0 1 6 30 112 364 1026 2652 6288 13995 29372 58786 112632 208012 371384 
               643842 1086384 1789515 2882934
%C A051168 0 1 7 35 140 476 1428 3876 9690 22610 49742 104006 208012 400023 742900 
               1337220 2340135 3991995 6653325
%C A051168 0 1 7 40 168 612 1932 5537 14520 35530 81686 178296 371384 742900 1432613 
               2674440 4847208 8554275 14732005
%C A051168 ...
%C A051168 It is essentially symmetric: A(r,r+i)=A(r,r-i+1).
%C A051168 Some of the diagonals are:
%C A051168 A(r,r+1): A000108
%C A051168 A(r,r): A022553
%C A051168 A(r,r-1): A000108
%C A051168 A(r,r+2): A000150
%C A051168 A(r,r+3): A050181
%C A051168 A(r,r+4): A050182
%C A051168 A(r,r+5): A050183
%C A051168 A(r,r-2): A000150 (End)
%H A051168 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">
               Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A051168 <a href="Sindx_Lu.html#Lyndon">Index entries for sequences related to 
               Lyndon words</a>
%F A051168 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}.
%e A051168 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.
%e A051168 Rows: {1}; {1,1}; {0,1,0}; ...
%t A051168 Table[If[n===0,1,1/n Plus@@(MoebiusMu[ # ]Binomial[n/#,k/# ]&/@ Divisors[GCD[n,
               k]])],{n,0,12},{k,0,n}] - Wouter Meeussen (wouter.meeussen(AT)pandora.be), 
               Jul 20 2008
%o A051168 (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 */
%Y A051168 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.
%Y A051168 Cf. A047996, A052307, A052314, A092964.
%Y A051168 Sequence in context: A131026 A014604 A015199 this_sequence A163528 A160806 
               A133418
%Y A051168 Adjacent sequences: A051165 A051166 A051167 this_sequence A051169 A051170 
               A051171
%K A051168 nonn,tabl
%O A051168 0,18
%A A051168 Clark Kimberling (ck6(AT)evansville.edu)

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