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A034345 Number of binary [ n,4 ] codes without 0 columns.

%I A034345 #32 Jul 08 2025 20:16:09
%S A034345 0,0,0,1,4,11,27,63,134,276,544,1048,1956,3577,6395,11217,19307,32685,
%T A034345 54413,89225,144144,229647,360975,560259,858967,1301757,1950955,
%U A034345 2893102,4246868,6174084,8892966,12696295,17973092,25237467,35163431,48629902,66774760,91063984
%N A034345 Number of binary [ n,4 ] codes without 0 columns.
%C A034345 "We say that the sequence (a_n) is quasi-polynomial in n if there exist polynomials P_0, ..., P_{s-1} and an integer n_0 such that, for all n >= n_0, a_n = P_i(n) where i == n (mod s)." [This is from the abstract of Lisonek (2007), and he states that the condition "n >= n_0" makes his definition broader than the one in Stanley's book. From Section 5 of his paper, we conclude that (a(n): n >= 1) is a quasi-polynomial in n.] - _Petros Hadjicostas_, Oct 02 2019
%H A034345 Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>.
%H A034345 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A034345 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=4.]
%H A034345 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,4,2}.]
%H A034345 Petros Hadjicostas, <a href="/A034253/a034253.txt">Generating function for a(n)</a>.
%H A034345 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 A034345 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 A034345 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 A034345 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.
%H A034345 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.
%o A034345 (Sage) # Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
%o A034345 def A034253col(k, length):
%o A034345     G1 = PSL(k, GF(2))
%o A034345     G2 = PSL(k-1, GF(2))
%o A034345     D1 = G1.cycle_index()
%o A034345     D2 = G2.cycle_index()
%o A034345     f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
%o A034345     f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
%o A034345     f = f1 - f2
%o A034345     return f.taylor(x, 0, length).list()
%o A034345 # For instance the Taylor expansion for column k = 4 (this sequence) gives
%o A034345 print(A034253col(4, 30)) #
%Y A034345 Cf. A034254, A034344, A034346, A034347, A034348, A034349, A253186.
%Y A034345 Column k=4 of A034253 and first differences of A034358.
%K A034345 nonn
%O A034345 1,5
%A A034345 _N. J. A. Sloane_
%E A034345 More terms by _Petros Hadjicostas_, Oct 02 2019