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A245141 Number of endofunctions f on [2n] that are self-inverse on [n].

%I A245141 #16 Dec 05 2021 04:31:28
%S A245141 1,3,50,1626,83736,6026120,571350096,67996818960,9862902275456,
%T A245141 1700092943088768,342087177215788800,79115601821198404352,
%U A245141 20779757607847901690880,6133520505473954148381696,2017134796016735182500521984,733523863838078950241395968000
%N A245141 Number of endofunctions f on [2n] that are self-inverse on [n].
%C A245141 a(n) counts endofunctions f:{1,...,2n}-> {1,...,2n} with f(f(i))=i for all i in {1,...,n}.
%H A245141 Alois P. Heinz, <a href="/A245141/b245141.txt">Table of n, a(n) for n = 0..200</a>
%F A245141 a(n) = Sum_{i=0..n} C(n,i)^2 * i! * A000085(n-i) * (2*n)^(n-i).
%F A245141 a(n) = A245348(2n,n).
%e A245141 a(1) = 3: (1,1), (1,2), (2,1).
%p A245141 g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
%p A245141 a:= n-> add(binomial(n, i)^2*i!*g(n-i)*(2*n)^(n-i), i=0..n):
%p A245141 seq(a(n), n=0..20);
%t A245141 Join[{1}, Table[n! * Sum[Binomial[n,k] * 2^k * n^k* Sum[1/((k - 2*j)!*2^j*j!), {j, 0, Floor[k/2]}], {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Dec 05 2021 *)
%Y A245141 Cf. A000085, A245348.
%Y A245141 Column k=2 of A246070.
%K A245141 nonn
%O A245141 0,2
%A A245141 _Alois P. Heinz_, Jul 21 2014