A034348 - OEIS

%I #28 Oct 09 2025 11:22:15

%S 0,0,0,0,0,0,1,7,35,170,847,4408,24297,143270,901491,5985278,41175203,

%T 287813284,2009864185,13848061942,93369988436,613030637339,

%U 3908996099141,24179747870890,145056691643428,844229016035010,4769751989333029,26181645303024760,139750488576152520

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

%C To find the g.f., modify the SageMath program below (cf. function f). It is very complicated to write it here. - _Petros Hadjicostas_, Oct 05 2019

%H Discrete algorithms at the University of Bayreuth, <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a>.

%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>.

%H 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 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 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 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>.

%o (SageMath) # Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):

%o def A034253col(k, length):

%o     G1 = PSL(k, GF(2))

%o     G2 = PSL(k-1, GF(2))

%o     D1 = G1.cycle_index()

%o     D2 = G2.cycle_index()

%o     f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)

%o     f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)

%o     f = f1 - f2

%o     return f.taylor(x, 0, length).list()

%o # For instance the Taylor expansion for column k = 7 (this sequence) gives

%o print(A034253col(7, 30)) #

%Y Cf. A034254, A034344, A034345, A034346, A034347, A034349, A253186.

%Y Column k=7 of A034253 and first differences of A034361.

%K nonn

%O 1,8

%A _N. J. A. Sloane_

%E More terms from _Petros Hadjicostas_, Oct 05 2019