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A152875 Number of permutations of {1,2,...,n} with all odd entries preceding all even entries or all even entries preceding all odd entries.

%I A152875 #24 Aug 16 2023 12:28:21
%S A152875 1,1,2,4,8,24,72,288,1152,5760,28800,172800,1036800,7257600,50803200,
%T A152875 406425600,3251404800,29262643200,263363788800,2633637888000,
%U A152875 26336378880000,289700167680000,3186701844480000,38240422133760000,458885065605120000,5965505852866560000
%N A152875 Number of permutations of {1,2,...,n} with all odd entries preceding all even entries or all even entries preceding all odd entries.
%C A152875 a(n) = A152874(n,1).
%H A152875 Alois P. Heinz, <a href="/A152875/b152875.txt">Table of n, a(n) for n = 0..506</a>
%F A152875 a(2n) = 2n!^2; a(2n+1) = 2n!(n+1)! (for n>=2).
%F A152875 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).
%F A152875 D-finite with recurrence 4*a(n) -2*a(n-1) -n*(n-1)*a(n-2)=0. - _R. J. Mathar_, Jul 22 2022
%e A152875 a(4)=8 because we have 1324, 1342, 3124, 3142, 2413, 2431, 4213 and 4231.
%p A152875 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);
%p A152875 # second Maple program:
%p A152875 a:= n-> (h-> 2^signum(h)*h!*(n-h)!)(iquo(n, 2)):
%p A152875 seq(a(n), n=0..27);  # _Alois P. Heinz_, May 23 2023
%p A152875 # third Maple program:
%p A152875 a:= proc(n) option remember; `if`(n<4, n*(n-1)/2+1,
%p A152875        n*(n-1)*a(n-2)/4 +a(n-1)/2)
%p A152875     end:
%p A152875 seq(a(n), n=0..27);  # _Alois P. Heinz_, May 23 2023
%t A152875 a[n_] := Which[n<2, 1, EvenQ[n], 2(n/2)!^2, True, 2((n-1)/2)!*((n+1)/2)!];
%t A152875 Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Aug 16 2023 *)
%Y A152875 Cf. A124419, A152874.
%K A152875 nonn,easy
%O A152875 0,3
%A A152875 _Emeric Deutsch_, Dec 15 2008
%E A152875 a(0)=a(1)=1 prepended by _Alois P. Heinz_, May 23 2023