A283838 - OEIS

%I #22 Apr 05 2017 05:31:36

%S 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 162,57,15,3,304,299,111,31,7,1,492,551,215,61,15,3,797,1015,416,121,

%U 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 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 Alois P. Heinz, <a href="/A283838/b283838.txt">Rows n = 8..290, flattened</a>

%H 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 Triangle begins:

%e      1,

%e      3,

%e      5,     1,

%e      9,     3,

%e     16,     7,    1,

%e     26,    13,    3,

%e     43,    25,    7,    1,

%e     71,    47,   15,    3,

%e    115,    88,   29,    7,   1,

%e    187,   162,   57,   15,   3,

%e    304,   299,  111,   31,   7,   1,

%e    492,   551,  215,   61,  15,   3,

%e    797,  1015,  416,  121,  31,   7,  1,

%e   1291,  1867,  802,  239,  63,  15,  3,

%e   2089,  3435, 1547,  471, 125,  31,  7, 1,

%e   3381,  6319, 2983,  927, 249,  63, 15, 3,

%e   5472, 11624, 5751, 1824, 495, 127, 31, 7, 1,

%e   ...

%t gf[k_] := x^(2k)(x-x^k)^2 / ((1-x)(1-x^k)(1-2x+x^k));

%t T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];

%t Table[T[n, k], {n, 8, 24}, {k, 3, Floor[n/2]-1}] // Flatten (* _Jean-François Alcover_, Apr 05 2017 *)

%Y Cf. A094686, A283834-A283837.

%Y For row sums see A283839.

%K nonn,tabf

%O 8,2

%A _N. J. A. Sloane_, Mar 25 2017