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

def A034253col(k, length):

G1 = PSL(k, GF(2))

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

D1 = G1.cycle_index()

D2 = G2.cycle_index()

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

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

f = f1 - f2

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

# For instance the Taylor expansion for column k = 8 (current sequence) gives

print(A034253col(8, 30)) #

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>. [This is a lower triangular array whose lower triangle contains T(n,See column k). In the papers, the notation S_{nk2} is used=8.]

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_{nk2} = T(n,k)8,2}.]

Petros Hadjicostas, <a href="/A034253/a034253.txt">Generating function for column k = 4</a>. [Cf. A034345.]

Petros Hadjicostas, <a href="/A034253/a034253_1.txt">Generating function for column k = 5</a>. [Cf. A034346.]

Petros Hadjicostas, <a href="/A034253/a034253_2.txt">Generating function for column k = 6</a>. [Cf. A034347.]

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

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

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

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

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>. [This is a lower triangular array whose lower triangle contains T(n,k). In the papers, the notation S_{nk2} is used.]

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 S_{nk2} = T(n,k).]

Petros Hadjicostas, <a href="/A034253/a034253.txt">Generating function for column k = 4</a>. [Cf. A034345.]

Petros Hadjicostas, <a href="/A034253/a034253_1.txt">Generating function for column k = 5</a>. [Cf. A034346.]

Petros Hadjicostas, <a href="/A034253/a034253_2.txt">Generating function for column k = 6</a>. [Cf. A034347.]

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.]

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.

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.

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

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

Column k=8 of A034253.

approved

editing

_N. J. A. Sloane (njas(AT)research.att.com)_.

nonn,new

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Binary Number of binary [ n,8 ] codes without 0 columns.

nonn,new

nonn

H. Fripertinger, <a href="http://bedvgmwww.kfunigrazmathe2.acuni-bayreuth.at:8001de/frib/codes/tables.html">Web siteIsometry Classes of Codes</a>

nonn,new

nonn

Binary [ n,8 ] codes without 0 columns.

0, 0, 0, 0, 0, 0, 0, 1, 8, 47, 277, 1775, 12616, 102445, 957357, 10174566, 119235347, 1482297912, 18884450721, 240477821389, 3012879828566, 36800049400028, 436068618826236, 5001537857507095, 55482177298724426

1,9

H. Fripertinger and A. Kerber, in AAECC-11, Lect. Notes Comp. Sci. 948 (1995), 194-204.

<a href="http://bedvgm.kfunigraz.ac.at:8001/frib/codes/tables.html">Web site</a>

Cf. A034253, A034344-.

nonn

njas

approved