A002525 Number of permutations according to distance. (Formerly M1203 N0463)

0, 1, 2, 4, 10, 24, 55, 128, 300, 700, 1632, 3809, 8890, 20744, 48406, 112960, 263599, 615120, 1435416, 3349624, 7816528, 18240289, 42564706, 99327052, 231785058, 540883000, 1262179815, 2945365040, 6873169028, 16038912628

Offset

0,3

References

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (2,0,2,0,-1).

Formula

G.f.: x/(1 - 2*x - 2*x^3 + x^5). - Simon Plouffe

a(n) = Sum_{k=0..n-1} A002524(k). - Sean A. Irvine, Mar 10 2014

Maple

A002525:=z/(1-2*z-2*z**3+z**5); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A002524.

Sequence in context: A132732 A275447 A095214 * A159328 A190587 A190794

Adjacent sequences: A002522 A002523 A002524 * A002526 A002527 A002528

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Mar 10 2014

Status

approved