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A272261 Number of one-to-one functions f from [n] to [2n] where f(x) may not be equal to x or to 2n+1-x.

%I A272261 #34 May 10 2022 14:38:47
%S A272261 1,0,4,40,576,10528,233920,6124032,184656640,6302821888,240245858304,
%T A272261 10115537336320,466275700903936,23354247194542080,1262994451308888064,
%U A272261 73347095164693676032,4552571878016243466240,300763132329730843475968,21071629550593224017182720
%N A272261 Number of one-to-one functions f from [n] to [2n] where f(x) may not be equal to x or to 2n+1-x.
%H A272261 Marko R. Riedel, Brian M. Scott et al. <a href="http://math.stackexchange.com/questions/1755873/">Enumerating a type of function by inclusion-exclusion</a>
%F A272261 a(n) = Sum_{q=0..n} C(n,q) (-1)^q 2^q C(2n-q,n-q) (n-q)!.
%F A272261 a(n) = abs(A000806(n)) * 2^n.
%F A272261 E.g.f.: exp(-1+sqrt(1-4*x))/sqrt(1-4*x). - _Benedict W. J. Irwin_, Jul 14 2016
%F A272261 a(n) ~ 2^(2*n+1/2) * n^n / exp(n+1). - _Vaclav Kotesovec_, Jul 16 2016
%F A272261 Conjecture: Alternating sign g.f. is Sum_{k>=0} HermiteH[k,sqrt(x)]x^(k/2). - _Benedict W. J. Irwin_, Nov 30 2016
%F A272261 Conjecture D-finite with recurrence: a(n) + 2*(-2*n+1)*a(n-1) - 4*a(n-2)=0. - _R. J. Mathar_, Jan 27 2020
%F A272261 a(n) = KummerU(-n, -2*n, -2). - _Peter Luschny_, May 10 2022
%p A272261 a := n -> add(binomial(n,q)*(-1)^q*2^q*binomial(2*n-q,n-q)*(n-q)!, q=0..n): seq(a(n), n=0..20);
%p A272261 seq(simplify(KummerU(-n, -2*n, -2)), n = 0..18); # _Peter Luschny_, May 10 2022
%t A272261 Table[CoefficientList[Series[E^(-1 + Sqrt[1 - 4 x])/Sqrt[1 - 4 x], {x, 0, 20}],x][[n]] (n - 1)!, {n, 1, 20}] (* _Benedict W. J. Irwin_, Jul 14 2016 *)
%Y A272261 Cf. A000806.
%K A272261 nonn
%O A272261 0,3
%A A272261 _Marko Riedel_, Apr 23 2016