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A000270 Number of discordant permutations.
(Formerly M3019 N1221)
1
1, 1, 0, 3, 16, 95, 672, 5397, 48704, 487917, 5373920, 64547175, 839703696, 11762247419, 176509466560, 2825125339305, 48040633506048, 864932233294681, 16436901752820288, 328791893988472843, 6905593482159150480, 151941269284478380119, 3495011687269591273312 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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).

J. Touchard, Permutations discordant with two given permutations, Scripta Math., 19 (1953), 109-119.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..175

FORMULA

G.f.: 1+(1-x)/(1+x)*Sum_{n>=0} n*n!*(x/(1+x)^2)^n. - Vladeta Jovovic, Jun 29 2007

Recurrence: (n-3)*a(n) = (n-3)*n*a(n-1) + (n-3)*n*a(n-2) + n*a(n-3). - Vaclav Kotesovec, Mar 15 2014

a(n) ~ (n+1)! / exp(2). - Vaclav Kotesovec, Mar 15 2014

MAPLE

a:= n-> coeftayl(1+ (1-x)/(1+x) *add(k*k! *(x/ (1+x)^2)^k, k=0..n), x=0, n): seq(a(n), n=0..25); # Alois P. Heinz, Sep 24 2008

A000270 := proc(n) if n = 0 then 1 else n * add((-1)^(n-s)*s!*binomial(s+n-1, 2*s-1), s=1..n) fi end; seq(A000270(n), n=0..30);  # Mark van Hoeij, May 12 2013

MATHEMATICA

max = 20; f[x_] := 1+(1-x)/(1+x)*Sum[ n*n!*(x/(1+x)^2)^n, {n, 0, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 09 2011, after Vladeta Jovovic *)

a[n_] := -(-1)^n*n^2*HypergeometricPFQ[{2, 1-n, 1+n}, {3/2}, 1/4]; a[0]=1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, May 13 2013, after Mark van Hoeij *)

CROSSREFS

Sequence in context: A114174 A181067 A006347 * A157051 A000271 A157016

Adjacent sequences:  A000267 A000268 A000269 * A000271 A000272 A000273

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Alois P. Heinz, Sep 24 2008

STATUS

approved

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Last modified May 23 08:55 EDT 2017. Contains 286909 sequences.