%I #20 Dec 05 2019 17:39:44
%S 1,1,1,1,2,1,1,4,2,1,1,6,6,2,1,1,10,11,7,2,1,1,14,24,14,8,2,1,1,22,42,
%T 35,16,9,2,1,1,30,81,68,45,18,10,2,1,1,46,138,149,89,55,20,11,2,1,1,
%U 62,250,282,216,110,66,22,12,2,1,1,94,419,577,422,285,132,78,24,13,2,1
%N Triangle read by rows: T(n,k) (1 <= k <= n) = one-half of the number of binary vectors of length n and cuts-resistance k.
%C Cuts-resistance is defined in A319416.
%C This triangle summarizes the data shown in A319420.
%C Conjecture (Sloane): Sum_{i = 1..n} i * T(n,i) = A189391(n).
%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. See page 4.
%H N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%F T(n,k) = A329860(n,k)/2. - _Gus Wiseman_, Nov 25 2019
%e Triangle begins:
%e 1
%e 1 1
%e 1 2 1
%e 1 4 2 1
%e 1 6 6 2 1
%e 1 10 11 7 2 1
%e 1 14 24 14 8 2 1
%e 1 22 42 35 16 9 2 1
%e 1 30 81 68 45 18 10 2 1
%e 1 46 138 149 89 55 20 11 2 1
%e 1 62 250 282 216 110 66 22 12 2 1
%e 1 94 419 577 422 285 132 78 24 13 2 1
%e Lenormand gives first 15 rows.
%e For example, the "1,2,1" row here refers to the 8 vectors of length 3. There are 2 vectors of cuts-resistance 1, namely 010 and 101 (see A319416), 4 vectors of cuts-resistance 2 (100,011,001,110), and 2 of cuts-resistance 3 (000 and 111). Halving these counts we get 1,2,1
%t degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t Table[Length[Select[Tuples[{0,1},n],First[#]==1&°dep[#]==k&]],{n,8},{k,n}] (* _Gus Wiseman_, Nov 25 2019 *)
%Y Row sums are A000079.
%Y Column k = 2 appears to be A027383.
%Y The version for runs-resistance is A319411 or A329767.
%Y The version for compositions is A329861.
%Y The cuts-resistance of the binary expansion of n is A319416(n).
%Y Cf. A000975, A164707, A189391, A261983, A318921, A318928, A319420, A329738, A329860, A329865.
%K nonn,tabl
%O 1,5
%A _N. J. A. Sloane_, Sep 23 2018