A152873 Number of permutations of {1,2,...,n} (n>=2) having a single run of even entries. For example, the permutation 513284679 has a single run of even entries: 2846.

2, 6, 12, 48, 144, 720, 2880, 17280, 86400, 604800, 3628800, 29030400, 203212800, 1828915200, 14631321600, 146313216000, 1316818944000, 14485008384000, 144850083840000, 1738201006080000, 19120211066880000, 248562743869440000, 2982752926433280000, 41758540970065920000

Offset

2,1

Links

Table of n, a(n) for n=2..25.

Formula

a(n) = A152667(n,1).

a(2n) = (n+1)(n!)^2;

a(2n+1) = n!(n+2)!

E.g.f.: 24*sqrt(4-x^2)*arcsin(x/2)/[(2-x)^3*(2+x)^2] - x(6-8x-3x^2+2x^3)/ [(2+x)(2-x)^2].

G.f.: G(0)/x^2 -1/x^2 -2/x, where G(k) = 1 + x*(k+2)/(1 - x*(k+1)/ (x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013

D-finite with recurrence 4*a(n) -2*a(n-1) -(n+2)*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 24 2022

Sum_{n>=2} 1/a(n) = BesselI(1, 2) + BesselI(2, 2) - 3/2 = A096789 + A229020 - 3/2. - Amiram Eldar, Jan 22 2023

Example

a(4) = 12 because we have 2413, 2431, 4213, 4231, 1243, 1423 and their reverses.

Maple

ae := proc (n) options operator, arrow: factorial(n)^2*(n+1) end proc: ao := proc (n) options operator, arrow: factorial(n)*factorial(n+2) end proc: a := proc (n) if `mod`(n, 2) = 0 then ae((1/2)*n) else ao((1/2)*n-1/2) end if end proc; seq(a(n), n = 2 .. 23);
# second Maple program:
a:= n-> (h-> h!*(h+1+(n mod 2))!)(iquo(n, 2)):
seq(a(n), n=2..25);  # Alois P. Heinz, Sep 24 2024

Mathematica

a[n_] := If[OddQ[n], ((n - 1)/2)!*((n + 3)/2)!, (n/2 + 1) ((n/2)!)^2]; Array[a, 25, 2] (* Amiram Eldar, Jan 22 2023 *)

Crossrefs

Cf. A152667.

Cf. A096789, A229020.

Sequence in context: A292132 A208147 A369330 * A175810 A083001 A285079

Adjacent sequences: A152870 A152871 A152872 * A152874 A152875 A152876

Keyword

nonn

Author

Emeric Deutsch, Dec 14 2008

Status

approved