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A006167
Number of factorization patterns of polynomials of degree n over F_2.
(Formerly M2349)
5
1, 3, 4, 8, 11, 20, 27, 45, 61, 95, 128, 193, 257, 374, 497, 703, 927, 1287, 1683, 2297, 2987, 4013, 5186, 6887, 8843, 11614, 14836, 19294, 24514, 31622, 39968, 51167, 64377, 81839, 102509, 129528, 161539, 202959, 252124, 315110, 389949, 485062
OFFSET
1,2
COMMENTS
Let F_q(n) represent the number of factorization patterns of n with the property that there exists a monic polynomial V of degree n over the finite field F_q such that V factors over F_q into one of the F_q(n) factorization patterns. Sequence is for the q=2 case,
REFERENCES
R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
A. K. Agarwal and G. L. Mullen, Partitions with "d(a) copies of a", J. Combin. Theory, A48 (1988), 120-135.
R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82. (Annotated scanned copy)
FORMULA
Euler transform of sequence b(n) = sum_{d|n, A001037(d)>=n/d} 1. - Franklin T. Adams-Watters, Jun 19 2006
EXAMPLE
For n=3 there are 5 factorization patterns of cubic polynomials: 3, 2 + 1, 1^3, 1^2 + 1, 1 + 1 + 1. For example 1^2 + 1 corresponds to a cubic polynomial which factors as a linear of multiplicity 2 and a second distinct linear factor. For q=2 the pattern 1 + 1 + 1 is not allowed since over F_2 there are only two distinct monic irreducibles of degree 1. Thus a(3) = 4.
MATHEMATICA
A001037[n_] := Sum[ MoebiusMu[n/d]*2^d, {d, Divisors[n]}]/n; b[n_] := Sum[ nd = A001037[d]; If[nd >= n/d, 1, 0], {d, Divisors[n]}]; EulerTransform[ seq_List ] := With[{m = Length[seq]}, CoefficientList[ Series[ Times @@ (1/(1 - x^Range[m])^seq), {x, 0, m}], x]]; A006167 = Rest[ EulerTransform[ Table[ b[n], {n, 1, 42}]]] (* Jean-François Alcover, Mar 15 2012, after Franklin T. Adams-Watters *)
CROSSREFS
KEYWORD
nonn,nice
EXTENSIONS
Additional comments from Gary Mullen, Jun 03 2003
More terms from Franklin T. Adams-Watters, Jun 19 2006
STATUS
approved