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A329860 Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.

%I A329860 #8 Nov 24 2019 00:00:33
%S A329860 1,0,2,0,2,2,0,2,4,2,0,2,8,4,2,0,2,12,12,4,2,0,2,20,22,14,4,2,0,2,28,
%T A329860 48,28,16,4,2,0,2,44,84,70,32,18,4,2,0,2,60,162,136,90,36,20,4,2,0,2,
%U A329860 92,276,298,178,110,40,22,4,2,0,2,124,500,564,432,220,132,44,24,4,2
%N A329860 Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.
%C A329860 For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.
%H A329860 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%F A329860 For positive indices, T(n,k) = 2 * A319421(n,k).
%e A329860 Triangle begins:
%e A329860    1
%e A329860    0   2
%e A329860    0   2   2
%e A329860    0   2   4   2
%e A329860    0   2   8   4   2
%e A329860    0   2  12  12   4   2
%e A329860    0   2  20  22  14   4   2
%e A329860    0   2  28  48  28  16   4   2
%e A329860    0   2  44  84  70  32  18   4   2
%e A329860    0   2  60 162 136  90  36  20   4   2
%e A329860    0   2  92 276 298 178 110  40  22   4   2
%e A329860    0   2 124 500 564 432 220 132  44  24   4   2
%e A329860 Row n = 4 counts the following words:
%e A329860   0101  0010  0001  0000
%e A329860   1010  0011  0111  1111
%e A329860         0100  1000
%e A329860         0110  1110
%e A329860         1001
%e A329860         1011
%e A329860         1100
%e A329860         1101
%t A329860 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329860 Table[Length[Select[Tuples[{0,1},n],degdep[#]==k&]],{n,0,10},{k,0,n}]
%Y A329860 Column k = 2 appears to be 2 * A027383.
%Y A329860 The version for runs-resistance is A319411 or A329767.
%Y A329860 The cuts-resistance of the binary expansion of n is A319416(n).
%Y A329860 The version for compositions is A329861.
%Y A329860 Cf. A000975, A164707, A261983, A318928, A319420, A319421, A329738, A329865.
%K A329860 nonn,tabl
%O A329860 0,3
%A A329860 _Gus Wiseman_, Nov 23 2019