A257546 Number of permutations of length n such that numbers at odd positions are monotone and numbers at even positions are also monotone.
1, 1, 2, 6, 24, 40, 80, 140, 280, 504, 1008, 1848, 3696, 6864, 13728, 25740, 51480, 97240, 194480, 369512, 739024, 1410864, 2821728, 5408312, 10816624, 20801200, 41602400, 80233200, 160466400, 310235040, 620470080, 1202160780, 2404321560, 4667212440
Offset: 0
Links
- Robert Israel, Table of n, a(n) for n = 0..2993
- Ran Pan, Exercise C, Project P.
Programs
-
Magma
[1,1,2,6] cat [4*Binomial(n,Floor(n/2)): n in [4..40]]; // Vincenzo Librandi, Apr 30 2015
-
Maple
f:= gfun:-rectoproc({a(n+2)= 4*(1+n)*a(n)/(n+3) + 2*a(n+1)/(n+3), seq(a(n)=[1,1,2,6,24,40,80][n+1],n=0..5)},a(n),remember): map(f, [$0..100]); # Robert Israel, May 12 2015 -
Mathematica
Table[If[n <= 4, n!, 4 Binomial[n, Floor[n/2]]], {n, 31}] (* Michael De Vlieger, Apr 29 2015 *)
Formula
a(n) = 4*binomial(n,floor(n/2)) for n > 4; a(n) = n! for n <= 4.
From Robert Israel, Apr 30 2015: (Start)
G.f.: -3*(1+x)*(1+2*x^2) - 2/x + 2*(2+1/x)/sqrt(1-4*x^2).
a(n+2) = (4*(1+n)*a(n) + 2*a(n+1))/(n+3) for n >= 4. (End)
Extensions
More terms from Vincenzo Librandi, Apr 30 2015