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Triangle read by rows: T(n,k) = number of ternary Lyndon words of length n with exactly k 1's
3

%I #8 Aug 24 2015 14:32:25

%S 1,2,1,1,2,0,2,4,2,0,3,8,5,2,0,6,16,16,8,2,0,9,32,38,26,9,2,0,18,64,

%T 96,80,40,12,2,0,30,128,220,224,137,56,13,2,0,56,256,512,596,448,224,

%U 74,16,2,0,99,512,1144,1536,1336,806,332,96,17,2,0,186,1024,2560,3840,3840

%N Triangle read by rows: T(n,k) = number of ternary Lyndon words of length n with exactly k 1's

%C Sum of rows equal to number of ternary Lyndon words A027376 first column (k=0) is equal to the number of binary Lyndon words A001037 third through sixth column (k=2,3,4,5) equal to A124720, A124721, A124722, A124723 T(n+1,n-1) entry equal to A042948

%H Alois P. Heinz, <a href="/A123223/b123223.txt">Rows n = 0..140, flattened</a>

%F G.f. for columns (except for k=0) given by 1/k*sum_{d|k} mu(d) x^k/(1-2*x^d)^(k/d) T(0,0) = 1 and T(n,0) = 1/n*sum_{d|n} mu(d)*2^(n/d) T(n,n) = 0 if n>1, T(n,n-1) = 2

%e Triangle begins:

%e 1,

%e 2,1,

%e 1,2,0,

%e 2,4,2,0,

%e 3,8,5,2,0,

%e 6,16,16,8,2,0,

%e 9,32,38,26,9,2,0,

%e 18,64,96,80,40,12,2,0,

%e T(n,1) = 2^(n-1) because all words beginning with a 1 and consisting of the rest 2's or 3's are ternary Lyndon words with exactly one 1.

%Y Cf. A027376, A001037, A124720, A124721, A124722, A124723, A051168, A042948.

%K nonn,tabl

%O 0,2

%A _Mike Zabrocki_, Nov 05 2006