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A034346 Number of binary [ n,5 ] codes without 0 columns.

%I A034346 #25 Oct 06 2019 01:53:33
%S A034346 0,0,0,0,1,5,17,54,163,465,1283,3480,9256,24282,62812,160106,401824,
%T A034346 992033,2406329,5730955,13393760,30709772,69079030,152473837,
%U A034346 330344629,702839150,1469214076,3019246455,6103105779,12142291541,23790590387,45932253637,87434850942,164188881007
%N A034346 Number of binary [ n,5 ] codes without 0 columns.
%H A034346 Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>.
%H A034346 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.
%H A034346 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=5.]
%H A034346 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,5,2}.]
%H A034346 Petros Hadjicostas, <a href="/A034253/a034253_1.txt">Generating function for a(n)</a>.
%H A034346 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 A034346 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 A034346 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 A034346 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.
%H A034346 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.
%o A034346 (Sage) # Fripertinger's method to find the g.f. of column k >= 2 (for small k):
%o A034346 def A034253col(k, length):
%o A034346     G1 = PSL(k, GF(2))
%o A034346     G2 = PSL(k-1, GF(2))
%o A034346     D1 = G1.cycle_index()
%o A034346     D2 = G2.cycle_index()
%o A034346     f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
%o A034346     f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
%o A034346     f = f1 - f2
%o A034346     return f.taylor(x, 0, length).list()
%o A034346 # For instance the Taylor expansion for column k = 5 gives a(n):
%o A034346 print(A034253col(5, 30)) # _Petros Hadjicostas_, Oct 04 2019
%Y A034346 Cf. A034254, A034344, A034345, A034347, A034348, A034349, A253186.
%Y A034346 Column k=5 of A034253 and first differences of A034359.
%K A034346 nonn
%O A034346 1,6
%A A034346 _N. J. A. Sloane_
%E A034346 More terms from _Petros Hadjicostas_, Oct 04 2019