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A034341
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Number of binary [ n,7 ] codes of dimension <= 7 without zero columns.
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2
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1, 2, 4, 8, 16, 36, 80, 193, 497, 1392, 4282, 14805, 57875, 258894, 1321280, 7570495, 47305333, 311742256, 2103025726, 14206632939, 94726167427, 618051904983, 3927156178649, 24243834619157, 145277300343585, 844969890205372, 4772180415241078, 26189419064610811, 139774809119967723
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OFFSET
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1,2
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COMMENTS
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To get the g.f. of this sequence (with a constant 1), modify the Sage program below (cf. function f). It is too complicated to write it here. - Petros Hadjicostas, Sep 30 2019
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LINKS
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Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used by Harald Fripertinger to compute T_{nk2} = A076832(n,k) using the cycle index of PGL_k(2). Here k = 7. That is, a(n) = T_{n,7,2} = A076832(n,7), but we start at n = 1 rather than at n = 7.]
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. [The notation for A076832(n,k) is T_{nk2}. Here k = 7.]
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PROG
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(Sage) # Fripertinger's method to find the g.f. of column k for small k:
def Tcol(k, length):
G = PSL(k, GF(2))
D = G.cycle_index()
f = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D)
return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 7 gives a(n):
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CROSSREFS
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Column k=7 of A076832 (starting at n=7).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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