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.

1, 0, 4, 40, 576, 10528, 233920, 6124032, 184656640, 6302821888, 240245858304, 10115537336320, 466275700903936, 23354247194542080, 1262994451308888064, 73347095164693676032, 4552571878016243466240, 300763132329730843475968, 21071629550593224017182720

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Marko R. Riedel, Brian M. Scott et al. Enumerating a type of function by inclusion-exclusion

Formula

a(n) = Sum_{q=0..n} C(n,q) (-1)^q 2^q C(2n-q,n-q) (n-q)!.

a(n) = abs(A000806(n)) * 2^n.

E.g.f.: exp(-1+sqrt(1-4*x))/sqrt(1-4*x). - Benedict W. J. Irwin, Jul 14 2016

a(n) ~ 2^(2*n+1/2) * n^n / exp(n+1). - Vaclav Kotesovec, Jul 16 2016

Conjecture: Alternating sign g.f. is Sum_{k>=0} HermiteH[k,sqrt(x)]x^(k/2). - Benedict W. J. Irwin, Nov 30 2016

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

a(n) = KummerU(-n, -2*n, -2). - Peter Luschny, May 10 2022

Maple

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);
seq(simplify(KummerU(-n, -2*n, -2)), n = 0..18); # Peter Luschny, May 10 2022

Mathematica

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 *)

Crossrefs

Cf. A000806.

Sequence in context: A276362 A302178 A379252 * A291817 A128573 A052675

Adjacent sequences: A272258 A272259 A272260 * A272262 A272263 A272264

Keyword

nonn

Author

Marko Riedel, Apr 23 2016

Status

approved