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A001471
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Number of degree-n permutations of order exactly 3.
(Formerly M1833 N0727)
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28
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0, 0, 0, 2, 8, 20, 80, 350, 1232, 5768, 31040, 142010, 776600, 4874012, 27027728, 168369110, 1191911840, 7678566800, 53474964992, 418199988338, 3044269834280, 23364756531620, 199008751634000, 1605461415071822
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OFFSET
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0,4
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COMMENTS
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a(n) is the number of non-symmetric permutation matrices A of dimension n such that A^2 is the transpose of A. - Torlach Rush, Jul 09 2020
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = a(n-1) + (1 + a(n-3))*(n-1)(n-2).
a(n) = Sum_{j=1..floor(n/3)} n!/(j!*(n-3*j)!*(3^j)).
E.g.f.: exp(x + x^3/3) - exp(x).
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MATHEMATICA
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a[n_] := HypergeometricPFQ[{1/3-n/3, 2/3-n/3, -n/3}, {}, -9] - 1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 19 2011 *)
nxt[{n_, a_, b_, c_}]:={n+1, b, c, c+(1+a)(n-1)(n-2)}; NestList[nxt, {3, 0, 0, 0}, 25][[;; , 2]] (* Harvey P. Dale, Mar 09 2024 *)
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PROG
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(PARI) first(n)=my(v=vector(n+1)); for(i=3, n, v[i+1]=v[i] + (1+v[i-2])*(i-1)*(i-2)); v \\ Charles R Greathouse IV, Jul 10 2020
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^3/3) )); [Factorial(n-1)*b[n]-1: n in [1..m]]; // G. C. Greubel, May 14 2019
(Sage) m = 30; T = taylor(exp(x + x^3/3) -exp(x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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