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A034345
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Number of binary [ n,4 ] codes without 0 columns.
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7
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0, 0, 0, 1, 4, 11, 27, 63, 134, 276, 544, 1048, 1956, 3577, 6395, 11217, 19307, 32685, 54413, 89225, 144144, 229647, 360975, 560259, 858967, 1301757, 1950955, 2893102, 4246868, 6174084, 8892966, 12696295, 17973092, 25237467, 35163431, 48629902, 66774760, 91063984
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OFFSET
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1,5
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COMMENTS
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"We say that the sequence (a_n) is quasi-polynomial in n if there exist polynomials P_0, ..., P_{s-1} and an integer n_0 such that, for all n >= n_0, a_n = P_i(n) where i == n (mod s)." [This is from the abstract of Lisonek (2007), and he states that the condition "n >= n_0" makes his definition broader than the one in Stanley's book. From Section 5 of his paper, we conclude that (a(n): n >= 1) is a quasi-polynomial in n.] - Petros Hadjicostas, Oct 02 2019
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LINKS
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Discrete algorithms at the University of Bayreuth, Symmetrica.
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,4,2}.]
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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 = 4 (this sequence) gives
print(A034253col(4, 30)) #
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CROSSREFS
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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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