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A034344 Number of binary [ n,3 ] codes without 0 columns.

%I A034344 #34 Oct 05 2019 01:27:41
%S A034344 0,0,1,3,6,12,21,34,54,82,120,174,244,337,458,613,808,1056,1361,1738,
%T A034344 2200,2759,3431,4240,5198,6333,7670,9235,11056,13175,15618,18432,
%U A034344 21660,25347,29543,34312,39702,45786,52633,60315,68910,78515,89206,101092,114276,128866,144978,162750,182298
%N A034344 Number of binary [ n,3 ] codes without 0 columns.
%C A034344 The g.f. function below was calculated in Sage (using Fripertinger's method) and compared with the one in Lisonek's (2007) Example 5.3 (p. 627). - _Petros Hadjicostas_, Oct 02 2019
%H A034344 Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>.
%H A034344 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A034344 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables_4.html">Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns</a>. [See column k = 3.]
%H A034344 H. Fripertinger and A. Kerber, <a href="https://doi.org/10.1007/3-540-60114-7_15">Isometry classes of indecomposable linear codes</a>. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,3,2}.]
%H A034344 Petr Lisonek, <a href="https://doi.org/10.1016/j.jcta.2006.06.013">Combinatorial families enumerated by quasi-polynomials</a>, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5. The g.f. is given in Example 5.3.]
%H A034344 David Slepian, <a href="https://archive.org/details/bstj39-5-1219">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.
%H A034344 David Slepian, <a href="https://doi.org/10.1002/j.1538-7305.1960.tb03958.x">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.
%H A034344 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.
%H A034344 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.
%F A034344 G.f.: (x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^7) = (-x^15 + 2*x^14 - x^13 + x^12 + x^9 - x^7 + x^4 + x^3)/((1 - x)^2*(-x^2 + 1)*(-x^3 + 1)^2*(-x^4 + 1)*(-x^7 + 1)). - _Petros Hadjicostas_, Oct 02 2019
%o A034344 (Sage) # Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
%o A034344 def A034253col(k, length):
%o A034344     G1 = PSL(k, GF(2))
%o A034344     G2 = PSL(k-1, GF(2))
%o A034344     D1 = G1.cycle_index()
%o A034344     D2 = G2.cycle_index()
%o A034344     f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
%o A034344     f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
%o A034344     f = f1 - f2
%o A034344     return f.taylor(x, 0, length).list()
%o A034344 # For instance the Taylor expansion for column k = 3 (this sequence) gives
%o A034344 print(A034253col(3, 30)) # _Petros Hadjicostas_, Oct 02 2019
%Y A034344 Cf. A034254, A034345, A034346, A034347, A034348, A034349, A253186.
%Y A034344 Column k=3 of A034253.
%Y A034344 First differences of A034357.
%K A034344 nonn
%O A034344 1,4
%A A034344 _N. J. A. Sloane_
%E A034344 More terms from _Petros Hadjicostas_, Oct 02 2019