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A336114 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,1); a(0)=a(1)=1.

%I A336114 #43 Feb 27 2025 08:59:16
%S A336114 1,1,6,64,930,17088,380870,9992064,301738626,10310669440,393355695942,
%T A336114 16573741095360,764401360062626,38304552622588224,2072335759298438790,
%U A336114 120390122318741003008,7474705606285243345410,493940966313183768532224,34613731176130328980714886
%N A336114 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,1); a(0)=a(1)=1.
%C A336114 Number of perfect matchings of a chord diagram with 2*n vertices, where neighboring vertices are joined by one chord, and any other pair of vertices is joined by two chords.
%H A336114 Dmitry Efimov, <a href="https://arxiv.org/abs/1904.08651">The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials</a>, arXiv:1904.08651 [math.CO], 2020.
%F A336114 a(n) = 2*n*Sum_{k=0..n} (-1)^(n-k)*(n+k-1)!/(k!*(n-k)!), n>=2.
%F A336114 D-finite with recurrence a(n+1) = (4*n+3)*a(n)-(4*n-7)*a(n-1)-a(n-2), n>=4.
%F A336114 D-finite with recurrence a(n+1) = (8*n^2*a(n)+(2*n+1)*a(n-1))/(2*n-1), n>=3.
%F A336114 a(n) = |A002119(n)|-|A002119(n-1)|, n>=2.
%F A336114 a(n) ~ (2*n)!/(sqrt(e)*n!).
%F A336114 a(n) = U(n,1+2*n,-1) for n >= 2, where U(a,b,c) is the confluent hypergeometric function of the second kind. - _Stefano Spezia_, Jul 22 2020
%e A336114 A symmetric 4x4 Toeplitz matrix A with the first row (0,1,2,1) has the form:
%e A336114 0 1 2 1
%e A336114 1 0 1 2
%e A336114 2 1 0 1
%e A336114 1 2 1 0.
%e A336114 Its hafnian equals Hf(A) = a12*a34+a13*a24+a14*a23 = 1*1+2*2+1*1 = 6 = a(2).
%t A336114 Join[{1,1},Table[2 HypergeometricU[n,1+2 n,-1],{n,2,16}]] (* _Stefano Spezia_, Jul 22 2020 *)
%Y A336114 Cf. A002119, A336286, A336400.
%K A336114 nonn,easy
%O A336114 0,3
%A A336114 _Dmitry Efimov_, Jul 21 2020