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A319421 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.

%I A319421 #20 Dec 05 2019 17:39:44
%S A319421 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 A319421 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 A319421 62,250,282,216,110,66,22,12,2,1,1,94,419,577,422,285,132,78,24,13,2,1
%N A319421 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 A319421 Cuts-resistance is defined in A319416.
%C A319421 This triangle summarizes the data shown in A319420.
%C A319421 Conjecture (Sloane): Sum_{i = 1..n} i * T(n,i) = A189391(n).
%H A319421 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 Apparently unpublished.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 A319421 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 A319421 T(n,k) = A329860(n,k)/2. - _Gus Wiseman_, Nov 25 2019
%e A319421 Triangle begins:
%e A319421    1
%e A319421    1   1
%e A319421    1   2   1
%e A319421    1   4   2   1
%e A319421    1   6   6   2   1
%e A319421    1  10  11   7   2   1
%e A319421    1  14  24  14   8   2   1
%e A319421    1  22  42  35  16   9   2   1
%e A319421    1  30  81  68  45  18  10   2   1
%e A319421    1  46 138 149  89  55  20  11   2   1
%e A319421    1  62 250 282 216 110  66  22  12   2   1
%e A319421    1  94 419 577 422 285 132  78  24  13   2   1
%e A319421 Lenormand gives first 15 rows.
%e A319421 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 A319421 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A319421 Table[Length[Select[Tuples[{0,1},n],First[#]==1&&degdep[#]==k&]],{n,8},{k,n}] (* _Gus Wiseman_, Nov 25 2019 *)
%Y A319421 Row sums are A000079.
%Y A319421 Column k = 2 appears to be A027383.
%Y A319421 The version for runs-resistance is A319411 or A329767.
%Y A319421 The version for compositions is A329861.
%Y A319421 The cuts-resistance of the binary expansion of n is A319416(n).
%Y A319421 Cf. A000975, A164707, A189391, A261983, A318921, A318928, A319420, A329738, A329860, A329865.
%K A319421 nonn,tabl
%O A319421 1,5
%A A319421 _N. J. A. Sloane_, Sep 23 2018