A187847 - OEIS

%I #39 Jan 07 2016 03:51:27

%S 1,0,1,4,14,78,504,3720,30960,256320,2656080,30078720,369774720,

%T 4906137600,69894316800,1064341555200,16190733081600,279499828608000,

%U 5100017213491200,98087346669312000,1983334021853184000,42063950934061056000,933754193111900160000

%N Number of permutations p of [n] with p(i) <> i^2.

%C Also number of permutations of [n] that have no square fixed points.

%H Alois P. Heinz, <a href="/A187847/b187847.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{j=0..floor(sqrt(n))} (-1)^j*C(floor(sqrt(n)),j)*(n-j)!.

%e a(3) = 4: (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1).

%p with(LinearAlgebra):

%p a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> `if`(j<>i^2, 1, 0)))):

%p seq(a(n), n=0..15);

%p # second Maple program:

%p a:= n->(p->add((-1)^(j)*binomial(p, j)*(n-j)!, j=0..p))(floor(sqrt(n))):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Nov 02 2014

%t a[n_] := With[{p = Floor[Sqrt[n]]}, Sum[(-1)^j*Binomial[p, j]*(n-j)!, {j, 0, p}]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* _Jean-François Alcover_, Jan 07 2016, adapted from Maple *)

%Y Cf. A161131, A161132, A247978.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Apr 11 2011