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A283838 Irregular triangle read by rows: T(n,k) (n >= 8, 3 <= k <= floor(n/2)-1) = number of binary vectors of length <= n that start with 1^k, 0, end with 1, 0^k, and the factor between 1^k and 0^k does not contain 0^k or 1^k.

%I A283838 #22 Apr 05 2017 05:31:36
%S A283838 1,3,5,1,9,3,16,7,1,26,13,3,43,25,7,1,71,47,15,3,115,88,29,7,1,187,
%T A283838 162,57,15,3,304,299,111,31,7,1,492,551,215,61,15,3,797,1015,416,121,
%U A283838 31,7,1,1291,1867,802,239,63,15,3,2089,3435,1547,471,125,31,7,1,3381,6319,2983,927,249,63,15,3
%N A283838 Irregular triangle read by rows: T(n,k) (n >= 8, 3 <= k <= floor(n/2)-1) = number of binary vectors of length <= n that start with 1^k, 0, end with 1, 0^k, and the factor between 1^k and 0^k does not contain 0^k or 1^k.
%H A283838 Alois P. Heinz, <a href="/A283838/b283838.txt">Rows n = 8..290, flattened</a>
%H A283838 Stefano Bilotta, <a href="http://arxiv.org/abs/1605.03785">Variable-length Non-overlapping Codes</a>, arXiv preprint arXiv:1605.03785 [cs.IT], 2016 [See Table 3].
%e A283838 Triangle begins:
%e A283838      1,
%e A283838      3,
%e A283838      5,     1,
%e A283838      9,     3,
%e A283838     16,     7,    1,
%e A283838     26,    13,    3,
%e A283838     43,    25,    7,    1,
%e A283838     71,    47,   15,    3,
%e A283838    115,    88,   29,    7,   1,
%e A283838    187,   162,   57,   15,   3,
%e A283838    304,   299,  111,   31,   7,   1,
%e A283838    492,   551,  215,   61,  15,   3,
%e A283838    797,  1015,  416,  121,  31,   7,  1,
%e A283838   1291,  1867,  802,  239,  63,  15,  3,
%e A283838   2089,  3435, 1547,  471, 125,  31,  7, 1,
%e A283838   3381,  6319, 2983,  927, 249,  63, 15, 3,
%e A283838   5472, 11624, 5751, 1824, 495, 127, 31, 7, 1,
%e A283838   ...
%t A283838 gf[k_] := x^(2k)(x-x^k)^2 / ((1-x)(1-x^k)(1-2x+x^k));
%t A283838 T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];
%t A283838 Table[T[n, k], {n, 8, 24}, {k, 3, Floor[n/2]-1}] // Flatten (* _Jean-François Alcover_, Apr 05 2017 *)
%Y A283838 Cf. A094686, A283834-A283837.
%Y A283838 For row sums see A283839.
%K A283838 nonn,tabf
%O A283838 8,2
%A A283838 _N. J. A. Sloane_, Mar 25 2017