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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. 12

%I #39 Jan 31 2023 05:27:09

%S 1,1,6,64,930,17088,380870,9992064,301738626,10310669440,393355695942,

%T 16573741095360,764401360062626,38304552622588224,2072335759298438790,

%U 120390122318741003008,7474705606285243345410

%N 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 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 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 a(n) = 2*n*Sum_{k=0..n} (-1)^(n-k)*(n+k-1)!/(k!*(n-k)!), n>=2.

%F D-finite with recurrence a(n+1) = (4*n+3)*a(n)-(4*n-7)*a(n-1)-a(n-2), n>=4.

%F D-finite with recurrence a(n+1) = (8*n^2*a(n)+(2*n+1)*a(n-1))/(2*n-1), n>=3.

%F a(n) = |A002119(n)|-|A002119(n-1)|, n>=2.

%F a(n) ~ (2*n)!/(sqrt(e)*n!).

%F 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 A symmetric 4x4 Toeplitz matrix A with the first row (0,1,2,1) has the form:

%e 0 1 2 1

%e 1 0 1 2

%e 2 1 0 1

%e 1 2 1 0.

%e Its hafnian equals Hf(A)=a12*a34+a13*a24+a14*a23=1*1+2*2+1*1=6.

%t Join[{1,1},Table[2 HypergeometricU[n,1+2 n,-1],{n,2,16}]] (* _Stefano Spezia_, Jul 22 2020 *)

%Y Cf. A002119, A336286, A336400.

%K nonn

%O 0,3

%A _Dmitry Efimov_, Jul 21 2020

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Last modified March 28 08:12 EDT 2024. Contains 371236 sequences. (Running on oeis4.)