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A336286 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,0); a(0)=a(1)=1.

%I A336286 #23 Feb 27 2025 08:59:21
%S A336286 1,1,5,57,859,16087,362781,9593105,291347603,9998539791,382732896853,
%T A336286 16169762600329,747423640472235,37523173542935207,2033249827596197549,
%U A336286 118278700627740322977,7352204062275501662371,486343759162888783503775,34112193002666850227154213
%N A336286 The hafnian of a symmetric Toeplitz matrix of order 2*n, n>=2 with the first row (0,1,2,...,2,0); a(0)=a(1)=1.
%C A336286 Number of perfect matchings of an arc diagram with 2*n vertices, where neighboring vertices are joined by one arc, the vertices 1 and 2*n are not adjacent if n>=2, and all other pairs of vertices are joined by two arcs.
%H A336286 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 A336286 a(n) = Sum_{k=0..n} (-1)^(n-k)*(n+k-1)!*(3*n-k)/(k!*(n-k)!), n>=2.
%F A336286 D-finite with recurrence a(n+1) = (4n+4)*a(n) - (8n-13)*a(n-1) - 2*a(n-2), n>=4.
%F A336286 D-finite with recurrence a(n+1) = ((32*n^2-12*n+2)*a(n) + (8*n+1)*a(n-1))/(8*n-7), n>=3.
%F A336286 a(n) = |A002119(n)| - 2*|A002119(n-1)|, n>=2.
%F A336286 a(n) ~ (2*n)!/sqrt(e)*n!.
%e A336286 A symmetric 4 X 4 Toeplitz matrix A with the first row (0,1,2,0) has the form:
%e A336286   0 1 2 0
%e A336286   1 0 1 2
%e A336286   2 1 0 1
%e A336286   0 2 1 0.
%e A336286 Its hafnian equals Hf(A) = a12*a34 + a13*a24 + a14*a23 = 1*1 + 2*2 + 0*1 = 5 = a(2).
%p A336286 [1,1,seq(add((-1)^(n-k-1)*(n+k-1)!*(-3*n+k)/(k!*(n-k)!),k=0..n),n=2..32)] # _Georg Fischer_, Jun 05 2021
%t A336286 Join[{1,1},RecurrenceTable[{a[n+1] == (4*n+4)*a[n]-(8*n-13)*a[n-1]-2*a[n-2], a[2]==5, a[3]==57, a[4]==859}, a[n], {n,2,32}]] (* _Georg Fischer_, Jun 05 2021 *)
%Y A336286 Cf. A002119, A336114, A336400.
%K A336286 nonn,easy
%O A336286 0,3
%A A336286 _Dmitry Efimov_, Jul 16 2020