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%I M1203 N0463 #55 Jan 22 2022 16:26:42
%S 0,1,2,4,10,24,55,128,300,700,1632,3809,8890,20744,48406,112960,
%T 263599,615120,1435416,3349624,7816528,18240289,42564706,99327052,
%U 231785058,540883000,1262179815,2945365040,6873169028,16038912628
%N Number of permutations according to distance.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002525/b002525.txt">Table of n, a(n) for n = 0..1000</a>
%H Torleiv Kløve, <a href="http://www.ii.uib.no/publikasjoner/texrap/pdf/2008-376.pdf"> Spheres of Permutations under the Infinity Norm - Permutations with limited displacement</a>, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
%H R. Lagrange, <a href="http://archive.numdam.org/article/ASENS_1962_3_79_3_199_0.pdf">Quelques résultats dans la métrique des permutations</a>, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,2,0,-1).
%F G.f.: x/(1 - 2*x - 2*x^3 + x^5). - _Simon Plouffe_
%F a(n) = Sum_{k=0..n-1} A002524(k). - _Sean A. Irvine_, Mar 10 2014
%p A002525:=z/(1-2*z-2*z**3+z**5); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t a[n_ /; n <= 2] := n; a[3]=4; a[4]=10; a[n_] := a[n] = 2*a[n-1] + 2*a[n-3] - a[n-5]; Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Mar 12 2014 *)
%o (PARI) a(n) = {z = x + x*O(x^n); gf = z/(1-2*z-2*z^3+z^5); polcoeff(gf, n);} \\ _Michel Marcus_, Mar 11 2014
%o (Magma) I:=[0,1,2,4,10]; [n le 5 select I[n] else 2*Self(n-1) +2*Self(n-3) -Self(n-5): n in [1..41]]; // _G. C. Greubel_, Jan 22 2022
%o (Sage) [( x/(1-2*x-2*x^3+x^5) ).series(x,n+1).list()[n] for n in (0..40)] # _G. C. Greubel_, Jan 22 2022
%Y Cf. A002524.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Mar 10 2014
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