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A336400 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,2,1,...,1,2); a(0)=1, a(1)=2.

%I A336400 #34 Sep 29 2023 02:10:53
%S A336400 1,2,9,44,303,2697,29438,380529,5683359,96290588,1824544857,
%T A336400 38229811083,877643031554,21906313145979,590665804363833,
%U A336400 17109084307014332,529833078045763263,17468521692479218209,610901505126064857854,22586913755160674065113
%N A336400 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,2,1,...,1,2); a(0)=1, a(1)=2.
%C A336400 Number of perfect matchings of a chord diagram with 2*n vertices, where neighboring vertices are joined by two chords, and any other pair of vertices is joined by one chord.
%H A336400 Georg Fischer, <a href="/A336400/b336400.txt">Table of n, a(n) for n = 0..200</a>
%H A336400 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 A336400 a(n) = n*Sum_{k=0..n} (n+k-1)!/(k!*(n-k)!*2^(k-1)), n>=2.
%F A336400 a(n) = A001515(n) + A001515(n-1), n>=2.
%F A336400 a(n) ~ (2*n)!*e/(n!*2^n).
%F A336400 a(n) = 2^(n+1)*U(n, 1+2*n, 2) for n >= 2, where U(a, b, c) is the confluent hypergeometric function. - _Stefano Spezia_, Jul 22 2020
%F A336400 a(n) = ((22*n^2-74*n+43)*a(n-1) + (10*n^2-54*n+77)*a(n-2) + 5*(n-4)*a(n-3)) / (11*n-29) for n >= 3. - _Alois P. Heinz_, Jul 28 2020
%F A336400 E.g.f.: - 1 - x + (1 + 1/sqrt(1-2*x))*exp(1 - sqrt(1-2* x)). - _G. C. Greubel_, Sep 28 2023
%e A336400 A symmetric 4x4 Toeplitz matrix A with the first row (0,2,1,2) has the form:
%e A336400 0 2 1 2
%e A336400 2 0 2 1
%e A336400 1 2 0 2
%e A336400 2 1 2 0.
%e A336400 Its hafnian equals Hf(A)=a12*a34+a13*a24+a14*a23=2*2+1*1+2*2=9.
%p A336400 a:= proc(n) option remember; `if`(n<3, [1, 2, 9][n+1],
%p A336400      ((22*n^2-74*n+43)*a(n-1)+(10*n^2-54*n+77)*a(n-2)
%p A336400       +5*(n-4)*a(n-3))/(11*n-29))
%p A336400     end:
%p A336400 seq(a(n), n=0..22);  # _Alois P. Heinz_, Jul 28 2020
%t A336400 Join[{1,2},Table[2^(n+1)*HypergeometricU[n,1+2 n,2],{n,2,19}]] (* _Stefano Spezia_, Jul 22 2020 *)
%t A336400 (* or *) Join[{1, 2}, Table[n*Sum[(n+k-1)!/(k!*(n-k)!*2^(k-1)),{k,0,n}],{n,2,200}]] (* _Georg Fischer_, Jul 28 2020 *)
%o A336400 (Magma) I:=[1,2,9]; [n le 3 select I[n] else ( (22*n^2-118*n+139)*Self(n-1) + (10*n^2-74*n+141)*Self(n-2) + 5*(n-5)*Self(n-3) )/(11*n-40): n in [1..30]]; // _G. C. Greubel_, Sep 28 2023
%o A336400 (SageMath)
%o A336400 def A336400(n): return n+1 if n<2 else (2/factorial(n-1))*sum(binomial(n,k)*gamma(n+k)/2^k for k in range(n+1))
%o A336400 [A336400(n) for n in range(31)] # _G. C. Greubel_, Sep 28 2023
%Y A336400 Cf. A001515, A336114, A336286.
%K A336400 nonn
%O A336400 0,2
%A A336400 _Dmitry Efimov_, Jul 22 2020
%E A336400 Incorrect recurrences removed and a(18) corrected by _Georg Fischer_, Jul 28 2020