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 A084575 Number of terms in polynomial expression for determinant of generic circulant matrix of order n. 1
 1, 2, 4, 10, 26, 68, 246, 810, 2704, 7492, 32066, 86500, 400024, 1366500, 4614524, 18784170, 68635478 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Define an n X n matrix A[i,j] by A[i,j]=x_(i+j), subscripts on x being interpreted mod n. This is a generic circulant matrix. If we expand det(A) we obtain a polynomial in the x_i. Define a(n) to be the number of terms in this polynomial after like terms have been combined. (Replacing det(A) with per(A), the permanent of A, we get sequence A003239). LINKS Hugh Thomas, The number of terms in the permanent ..., arXiv:math/0301048 [math.CO], 2003. FORMULA a(n) <= A003239(n), with = if n is a prime power. For other values of n little is known. EXAMPLE Example : for n=2 the matrix is x2,x1 x1,x2 and the determinant is (x_2)^2 - (x_1)^2 so a(2) = 2 and likewise for the permanent. MATHEMATICA Table[Clear[x]; r=Array[x, n]; m=Table[RotateRight[r, i], {i, 0, n-1}]; Length[Expand[Det[m]]], {n, 10}] (* T. D. Noe, Oct 22 2008 *) CROSSREFS Cf. A003239. Sequence in context: A055819 A052995 A113337 * A081881 A025565 A085455 Adjacent sequences:  A084572 A084573 A084574 * A084576 A084577 A084578 KEYWORD nonn,hard,more AUTHOR Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 13 2003 EXTENSIONS a(13) term added by T. D. Noe, Oct 22 2008 a(14) and a(15) from Roman Pearce, Aug 30 2014 a(16) and a(17) from Robert Israel, Aug 30 2014 STATUS approved

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Last modified February 19 09:57 EST 2020. Contains 332041 sequences. (Running on oeis4.)