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Search: id:A052182
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| A052182 |
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Determinant of n X n matrix whose rows are cyclic permutations of 1..n. |
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+0
17
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| 1, -3, 18, -160, 1875, -27216, 470596, -9437184, 215233605, -5500000000, 155624547606, -4829554409472, 163086595857367, -5952860799406080, 233543408203125000, -9799832789158199296, 437950726881001816329
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an odd number of cycles. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 30 2006
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REFERENCES
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P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..100
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FORMULA
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a(n) = (-1)^(n-1) * n^(n-2) * (n^2 + n)/2
E.g.f.[A052182] = E.g.f.[A000312] * E.g.f.[A000272], so A052182(unsigned) is "tree-like". E.g.f.: (T-T^2/2)/(1-T), where T=T(x) is Euler's tree function (see A000169). E.g.f. for signed sequence: (W+W^2/2)/(1+W), where W=W(x)=-T(-x) is the Lambert W function.- Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 13 2001
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EXAMPLE
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a(3) = 18 because this is the determinant of [(1,2,3), (3,1,2), (2,3,1) ]
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MATHEMATICA
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f[n_] := Module[{a = Table[i, {i, 1, n}], m = {}, k = 0}, While[k < n, m = Append[m, RotateLeft[a, k]]; k++ ]; Det[m]]; Table[ f[n], {n, 1, 20} ]
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PROGRAM
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(Mupad) (1+n)^(n-1)*binomial(n+2, n)*(-1)^(n) $ n=0..16 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2007
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CROSSREFS
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Cf. A000312, A070896, A060281, A060435.
Sequence in context: A075678 A089901 A067302 this_sequence A115415 A065058 A032031
Adjacent sequences: A052179 A052180 A052181 this_sequence A052183 A052184 A052185
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KEYWORD
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easy,sign,nice
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AUTHOR
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Henry M. Gunn High School Mathematical Circle (joshua.zucker(AT)stanfordalumni.org), Jan 26 2000
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 31 2000
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