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A034349
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Number of binary [ n,8 ] codes without 0 columns.
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7
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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, 595303034603214108, 6181562837200509792, 62170512250565592346
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OFFSET
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1,9
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COMMENTS
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To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - Petros Hadjicostas, Oct 07 2019
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LINKS
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Table of n, a(n) for n=1..28.
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=8.]
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,8,2}.]
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 = 8 (current sequence) gives
print(A034253col(8, 30)) # Petros Hadjicostas, Oct 07 2019
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CROSSREFS
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Cf. A034254, A034344, A034345, A034346, A034347, A034348, A253186.
Column k=8 of A034253 and first differences of A034362.
Sequence in context: A014524 A098891 A054488 * A296797 A024108 A247726
Adjacent sequences: A034346 A034347 A034348 * A034350 A034351 A034352
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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 07 2019
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STATUS
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approved
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