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A034348 Number of binary [ n,7 ] codes without 0 columns.

%I A034348 #23 Jul 08 2025 20:16:22
%S A034348 0,0,0,0,0,0,1,7,35,170,847,4408,24297,143270,901491,5985278,41175203,
%T A034348 287813284,2009864185,13848061942,93369988436,613030637339,
%U A034348 3908996099141,24179747870890,145056691643428,844229016035010,4769751989333029,26181645303024760,139750488576152520
%N A034348 Number of binary [ n,7 ] codes without 0 columns.
%C A034348 To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - _Petros Hadjicostas_, Oct 05 2019
%H A034348 Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>.
%H A034348 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A034348 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=7.]
%H A034348 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,7,2}.]
%H A034348 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.]
%H A034348 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 A034348 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 A034348 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.
%H A034348 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.
%o A034348 (Sage) # Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
%o A034348 def A034253col(k, length):
%o A034348     G1 = PSL(k, GF(2))
%o A034348     G2 = PSL(k-1, GF(2))
%o A034348     D1 = G1.cycle_index()
%o A034348     D2 = G2.cycle_index()
%o A034348     f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
%o A034348     f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
%o A034348     f = f1 - f2
%o A034348     return f.taylor(x, 0, length).list()
%o A034348 # For instance the Taylor expansion for column k = 7 (this sequence) gives
%o A034348 print(A034253col(7, 30)) #
%Y A034348 Cf. A034254, A034344, A034345, A034346, A034347, A034349, A253186.
%Y A034348 Column k=7 of A034253 and first differences of A034361.
%K A034348 nonn
%O A034348 1,8
%A A034348 _N. J. A. Sloane_
%E A034348 More terms from _Petros Hadjicostas_, Oct 05 2019