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A152876 Number of permutations of {1,2,...,n} having no consecutive triples of the form (odd, even, odd) or (even, odd, even).

%I A152876 #27 Nov 16 2019 15:48:44
%S A152876 1,1,2,4,16,60,288,1584,10368,74880,604800,5356800,51840000,544320000,
%T A152876 6147187200,74579097600,962415820800,13241346048000,192255565824000,
%U A152876 2957575348224000,47721518530560000,811595019755520000,14407079038894080000,268390402745794560000
%N A152876 Number of permutations of {1,2,...,n} having no consecutive triples of the form (odd, even, odd) or (even, odd, even).
%C A152876 Column 0 of A152877.
%H A152876 Alois P. Heinz, <a href="/A152876/b152876.txt">Table of n, a(n) for n = 0..460</a>
%H A152876 E. Munarini and N. Z. Salvi, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s49zagaglia.html">Binary strings without zigzags</a>, Séminaire Lotharingien de Combinatoire, B49h (2004), 1-15.
%F A152876 a(2n) = (n!)^2*Sum_{j=0..floor(n/2)} (binomial(n-j, j))^2*(2*n^2 - 6*j*n + 6*j^2)/(n-j)^2;
%F A152876 a(2n+1) = n!*(n+1)!*Sum_{j=1..floor(n/2)} binomial(n+1-j, j)*binomial(n-j, j)*(2*n^2 + 2*n - 6*j*n - 3*j + 6*j^2)/((n+1-j)*(n-j)).
%F A152876 a(n) ~ 2 * 5^(-1/4) * ((1+sqrt(5))/4)^n * n!. - _Vaclav Kotesovec_, Sep 03 2014
%e A152876 a(3) = 4 because we have 132, 213, 231 and 312.
%p A152876 ae := proc (n) options operator, arrow: 2*(sum((n^2-3*n*j+3*j^2)*binomial(n-j, j)^2/(n-j)^2, j = 0 .. floor((1/2)*n))) end proc: ao := proc (n) options operator, arrow: sum(binomial(n+1-k, k)*binomial(n-k, k)*(2*n^2+2*n-6*k*n-3*k+6*k^2)/((n+1-k)*(n-k)), k = 0 .. floor((1/2)*n)) end proc: a := proc (n) if `mod`(n, 2) = 0 then factorial((1/2)*n)^2*ae((1/2)*n) else factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*ao((1/2)*n-1/2) end if end proc: 1, 1, seq(a(n), n = 2 .. 22);
%p A152876 # second Maple program:
%p A152876 b:= proc(o, u, t) option remember; `if`(u+o=0, 1,
%p A152876       `if`(t=4, 0, o*b(o-1, u, `if`(t=3, 5, 2)))+
%p A152876       `if`(t=5, 0, u*b(o, u-1, `if`(t=2, 4, 3))))
%p A152876     end:
%p A152876 a:= n-> b(ceil(n/2), floor(n/2), 1):
%p A152876 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 11 2013
%t A152876 b[o_, u_, t_] := b[o, u, t] = If[u+o == 0, 1, If[t==4, 0, o*b[o-1, u, If[t==3, 5, 2]]] + If[t==5, 0, u*b[o, u-1, If[t==2, 4, 3]]]]; a[n_] := b[Ceiling[n/2], Floor[n/2], 1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 29 2015, after _Alois P. Heinz_ *)
%Y A152876 Cf. A152877.
%K A152876 nonn
%O A152876 0,3
%A A152876 _Emeric Deutsch_, Dec 17 2008