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A336114
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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.
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12
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1, 1, 6, 64, 930, 17088, 380870, 9992064, 301738626, 10310669440, 393355695942, 16573741095360, 764401360062626, 38304552622588224, 2072335759298438790, 120390122318741003008, 7474705606285243345410
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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Join[{1, 1}, Table[2 HypergeometricU[n, 1+2 n, -1], {n, 2, 16}]] (* Stefano Spezia, Jul 22 2020 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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