A152875 Number of permutations of {1,2,...,n} with all odd entries preceding all even entries or all even entries preceding all odd entries.
1, 1, 2, 4, 8, 24, 72, 288, 1152, 5760, 28800, 172800, 1036800, 7257600, 50803200, 406425600, 3251404800, 29262643200, 263363788800, 2633637888000, 26336378880000, 289700167680000, 3186701844480000, 38240422133760000, 458885065605120000, 5965505852866560000
Offset: 0
Examples
a(4)=8 because we have 1324, 1342, 3124, 3142, 2413, 2431, 4213 and 4231.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..506
Programs
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Maple
a := proc (n) if `mod`(n, 2) = 0 then 2*factorial((1/2)*n)^2 else 2*factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2) end if end proc: seq(a(n), n = 2 .. 25); # second Maple program: a:= n-> (h-> 2^signum(h)*h!*(n-h)!)(iquo(n, 2)): seq(a(n), n=0..27); # Alois P. Heinz, May 23 2023 # third Maple program: a:= proc(n) option remember; `if`(n<4, n*(n-1)/2+1, n*(n-1)*a(n-2)/4 +a(n-1)/2) end: seq(a(n), n=0..27); # Alois P. Heinz, May 23 2023
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Mathematica
a[n_] := Which[n<2, 1, EvenQ[n], 2(n/2)!^2, True, 2((n-1)/2)!*((n+1)/2)!]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 16 2023 *)
Formula
a(2n) = 2n!^2; a(2n+1) = 2n!(n+1)! (for n>=2).
E.g.f.: 1+x+2*(4*sqrt(4-x^2)*arcsin(x/2) - 4x + 4x^2 + x^3 - x^4)/((2+x)*(2-x)^2).
D-finite with recurrence 4*a(n) -2*a(n-1) -n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 22 2022
Extensions
a(0)=a(1)=1 prepended by Alois P. Heinz, May 23 2023
Comments