A002525 Number of permutations according to distance.

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

Author

N. J. A. Sloane

Keywords

• nonn

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).

Crossrefs

Cf. A002524.

Programs

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

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 *)

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

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

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

Extensions

More terms from Sean A. Irvine, Mar 10 2014