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 A034345 Number of binary [ n,4 ] codes without 0 columns. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS "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 LINKS 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=4.] 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}.] 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. PROG (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*prod(1/(1-x^j) for j in i) for i in D1)     f2 = sum(i*prod(1/(1-x^j) for j in i) 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)) # CROSSREFS Cf. A034254, A034344, A034346, A034347, A034348, A034349, A253186. Column k=4 of A034253 and first differences of A034358. Sequence in context: A295056 A160399 A119706 * A036890 A000253 A276691 Adjacent sequences:  A034342 A034343 A034344 * A034346 A034347 A034348 KEYWORD nonn AUTHOR EXTENSIONS More terms by Petros Hadjicostas, Oct 02 2019 STATUS approved

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Last modified April 17 11:26 EDT 2021. Contains 343064 sequences. (Running on oeis4.)