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A285672 Number of permutations p of [n] avoiding consecutive odd sums i+p(i), (i+1)+p(i+1) for all i in [n-1].

%I A285672 #8 Aug 30 2021 09:39:27
%S A285672 1,1,1,2,8,36,180,1008,6336,46080,374400,3369600,32659200,344736000,
%T A285672 3886444800,47348582400,611264102400,8442272563200,122595843686400,
%U A285672 1890952003584000,30510694932480000,520011800985600000,9231875243458560000,172292221923655680000
%N A285672 Number of permutations p of [n] avoiding consecutive odd sums i+p(i), (i+1)+p(i+1) for all i in [n-1].
%H A285672 Alois P. Heinz, <a href="/A285672/b285672.txt">Table of n, a(n) for n = 0..466</a>
%e A285672 a(0) = 1: the empty permutation.
%e A285672 a(1) = 1: 1.
%e A285672 a(2) = 1: 12.
%e A285672 a(3) = 2: 123, 321.
%e A285672 a(4) = 8: 1234, 1432, 2413, 2431, 3214, 3412, 4213, 4231.
%e A285672 a(5) = 36: 12345, 12543, 13524, 13542, 14325, 14523, 15324, 15342, 24135, 24153, 24315, 24351, 24513, 24531, 31524, 31542, 32145, 32541, 34125, 34521, 35124, 35142, 42135, 42153, 42315, 42351, 42513, 42531, 51324, 51342, 52143, 52341, 53124, 53142, 54123, 54321.
%p A285672 b:= proc(n, i, j, p, t) option remember; `if`(n=0, 1,
%p A285672      `if`(i=0 or t=1 and p=1, 0, i*b(n-1, i-1, j, 1-p, p))+
%p A285672      `if`(j=0 or t=1 and p=0, 0, j*b(n-1, i, j-1, 1-p, 1-p)))
%p A285672     end:
%p A285672 a:= n-> b(n, floor(n/2), ceil(n/2), 1, 0):
%p A285672 seq(a(n), n=0..25);
%t A285672 b[n_, i_, j_, p_, t_] := b[n, i, j, p, t] =
%t A285672      If[n==0, 1, If[i==0 || t ==1 && p==1, 0, i*b[n-1, i-1, j, 1-p, p]] +
%t A285672      If[j==0 || t==1 && p==0, 0, j*b[n-1, i, j-1, 1-p, 1-p]]];
%t A285672 a[n_] := b[n, Floor[n/2], Ceiling[n/2], 1, 0];
%t A285672 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 30 2021, after _Alois P. Heinz_ *)
%Y A285672 Cf. A199660, A231601.
%K A285672 nonn
%O A285672 0,4
%A A285672 _Alois P. Heinz_, Apr 24 2017