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.

1, 1, 6, 64, 930, 17088, 380870, 9992064, 301738626, 10310669440, 393355695942, 16573741095360, 764401360062626, 38304552622588224, 2072335759298438790, 120390122318741003008, 7474705606285243345410, 493940966313183768532224, 34613731176130328980714886

Offset: 0

Author

Dmitry Efimov, Jul 21 2020

Keywords

• nonn
• easy

Comments

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.

Examples

			A symmetric 4x4 Toeplitz matrix A with the first row (0,1,2,1) has the form:
0 1 2 1
1 0 1 2
2 1 0 1
1 2 1 0.
Its hafnian equals Hf(A) = a12*a34+a13*a24+a14*a23 = 1*1+2*2+1*1 = 6 = a(2).
		

Links

• Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2020.

Crossrefs

Cf. A002119, A336286, A336400.

Programs

Mathematica

Join[{1,1},Table[2 HypergeometricU[n,1+2 n,-1],{n,2,16}]] (* Stefano Spezia, Jul 22 2020 *)

Formula

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

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

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

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

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

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