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%I #17 Feb 17 2024 12:51:17
%S 1,0,1,1,2,3,7,14,35,81,216,557,1583,4444,13389,40313,128110,409519,
%T 1366479,4603338,16064047,56708713,206238116,759535545,2870002519,
%U 10986716984,43019064953,170663829777,690840124506,2832976091771,11831091960887,50040503185030
%N Number of permutations of [n] having no isolated entries. An entry j of a permutation p is isolated if it is not preceded by j-1 and not followed by j+1. For example, the permutation 23178564 has 2 isolated entries: 1 and 4.
%C a(n) = A180196(n,0).
%C a(n) = n! - A184181(n).
%H Alois P. Heinz, <a href="/A180564/b180564.txt">Table of n, a(n) for n = 0..893</a>
%F a(n) = Sum_{j=1..floor(n/2)} binomial(n-j-1, j-1)*(d(j) + d(j-1)), where d(i) = A000166(i) are the derangement numbers; a(0)=1.
%e a(5)=3 because we have 12345, 34512, and 45123.
%p d[ -1] := 0: d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n) options operator, arrow: sum(binomial(n-j-1, j-1)*(d[j]+d[j-1]), j = 1 .. floor((1/2)*n)) end proc:a(0):=1: seq(a(n), n = 0 .. 32);
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<4, [1, 0, 1, 1][n+1],
%p (3*a(n-1)+(n-3)*a(n-2)-(n-3)*a(n-3)+(n-4)*a(n-4))/2)
%p end:
%p seq(a(n), n=0..31); # _Alois P. Heinz_, Feb 17 2024
%Y Cf. A000142, A000166, A180196, A184181.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Sep 09 2010
%E a(0)=1 prepended by _Alois P. Heinz_, Feb 17 2024
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