OFFSET
0,3
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.
LINKS
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 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) ~ (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
EXAMPLE
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.
MATHEMATICA
Join[{1, 1}, Table[2 HypergeometricU[n, 1+2 n, -1], {n, 2, 16}]] (* Stefano Spezia, Jul 22 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Dmitry Efimov, Jul 21 2020
STATUS
approved