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A034347
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Number of binary [ n,6 ] codes without 0 columns.
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7
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0, 0, 0, 0, 0, 1, 6, 25, 99, 385, 1472, 5676, 22101, 87404, 350097, 1413251, 5708158, 22903161, 90699398, 352749035, 1342638839, 4990325414, 18090636016, 63933709870, 220277491298, 740170023052, 2426954735273, 7770739437179, 24314436451415, 74406425640743, 222867051758565, 653898059035166
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OFFSET
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1,7
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LINKS
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Table of n, a(n) for n=1..32.
Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=6.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. 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,6,2}.]
Petros Hadjicostas, Generating function for a(n).
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.
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PROG
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(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 = 6 (this sequence gives
print(A034253col(6, 30)) # Petros Hadjicostas, Oct 05 2019
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CROSSREFS
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Cf. A034254, A034344, A034345, A034346, A034348, A034349, A253186.
First differences of A034360.
Column k = 6 of A034253.
Sequence in context: A214955 A286433 A034559 * A009121 A327504 A323824
Adjacent sequences: A034344 A034345 A034346 * A034348 A034349 A034350
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Petros Hadjicostas, Oct 05 2019
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STATUS
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approved
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