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%I #95 Nov 06 2023 07:14:35
%S 1,0,1,5,36,329,3655,47844,721315,12310199,234615096,4939227215,
%T 113836841041,2850860253240,77087063678521,2238375706930349,
%U 69466733978519340,2294640596998068569,80381887628910919255,2976424482866702081004,116160936719430292078411
%N Number of loopless linear chord diagrams with n chords.
%C See the signed version of these numbers, A000806, for much more information about these numbers.
%C From _Gus Wiseman_, Feb 27 2019: (Start)
%C Also the number of 2-uniform set partitions of {1..2n} containing no two successive vertices in the same block. For example, the a(3) = 5 set partitions are:
%C {{1,3},{2,5},{4,6}}
%C {{1,4},{2,5},{3,6}}
%C {{1,4},{2,6},{3,5}}
%C {{1,5},{2,4},{3,6}}
%C {{1,6},{2,4},{3,5}}
%C (End)
%C From _Gus Wiseman_, Jul 05 2020: (Start)
%C Also the number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal and where the first i appears before the first j for i < j. For example, the a(3) = 5 permutations are the following.
%C (1,2,3,1,2,3)
%C (1,2,3,1,3,2)
%C (1,2,3,2,1,3)
%C (1,2,3,2,3,1)
%C (1,2,1,3,2,3)
%C (End)
%H Seiichi Manyama, <a href="/A278990/b278990.txt">Table of n, a(n) for n = 0..404</a> (terms 0..200 from Gheorghe Coserea)
%H 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], 2019.
%H H. Eriksson and A. Martin, <a href="https://arxiv.org/abs/1702.04177">Enumeration of Carlitz multipermutations</a>, arXiv:1702.04177 [math.CO], 2017.
%H E. Krasko, I. Labutin, and A. Omelchenko, <a href="https://arxiv.org/abs/1709.03218">Enumeration of labelled and unlabelled Hamiltonian Cycles in complete k-partite graphs</a>, arXiv:1709.03218 [math.CO], 2017, Table 1.
%H E. Krasko and A. Omelchenko, <a href="https://arxiv.org/abs/1601.05073">Enumeration of Chord Diagrams without Loops and Parallel Chords</a>, arXiv:1601.05073 [math.CO], 2016.
%H E. Krasko and A. Omelchenko, <a href="https://doi.org/10.37236/6037">Enumeration of Chord Diagrams without Loops and Parallel Chords</a>, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
%H Gus Wiseman, <a href="/A278990/a278990.png">The a(4) = 36 loopless linear chord diagrams</a>.
%H Donovan Young, <a href="https://arxiv.org/abs/2311.01569">Counting Bubbles in Linear Chord Diagrams</a>, arXiv:2311.01569 [math.CO], 2023.
%F From _Gheorghe Coserea_, Dec 09 2016: (Start)
%F D-finite with recurrence a(n) = (2*n-1)*a(n-1) + a(n-2), with a(0) = 1, a(1) = 0.
%F E.g.f. y satisfies: 0 = (1-2*x)*y'' - 3*y' - y.
%F a(n) - a(n-1) = A003436(n) for all n >= 2.
%F (End)
%F From _Vaclav Kotesovec_, Sep 15 2017: (Start)
%F a(n) = sqrt(2)*exp(-1)*(BesselK(1/2 + n, 1)/sqrt(Pi) - i*sqrt(Pi)*BesselI(1/2 + n, -1)), where i is the imaginary unit.
%F a(n) ~ 2^(n+1/2) * n^n / exp(n+1).
%F (End)
%F a(n) = A114938(n)/n! - _Gus Wiseman_, Jul 05 2020 (from _Alexander Burstein_'s formula at A114938).
%F From _G. C. Greubel_, Sep 26 2023: (Start)
%F a(n) = (-1)^n * (i/e)*Sqrt(2/Pi) * BesselK(n + 1/2, -1).
%F G.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * Erfi((1+x)/sqrt(2*x)).
%F E.g.f.: exp(-1 + sqrt(1-2*x))/sqrt(1-2*x).
%t RecurrenceTable[{a[n]== (2n-1)a[n-1] +a[n-2], a[0]==1, a[1]==0}, a, {n,0,20}] (* _Vaclav Kotesovec_, Sep 15 2017 *)
%t FullSimplify[Table[-I*(BesselI[1/2+n,-1] BesselK[3/2,1] - BesselI[3/2,-1] BesselK[1/2+ n,1]), {n,0,20}]] (* _Vaclav Kotesovec_, Sep 15 2017 *)
%t Table[(2 n-1)!! Hypergeometric1F1[-n,-2 n,-2], {n,0,20}] (* _Eric W. Weisstein_, Nov 14 2018 *)
%t Table[Sqrt[2/Pi]/E ((-1)^n Pi BesselI[1/2+n,1] +BesselK[1/2+n,1]), {n,0,20}] // FunctionExpand // FullSimplify (* _Eric W. Weisstein_, Nov 14 2018 *)
%t twouniflin[{}]:={{}};twouniflin[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@twouniflin[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+1&]}];
%t Table[Length[twouniflin[Range[n]]],{n,0,14,2}] (* _Gus Wiseman_, Feb 27 2019 *)
%o (PARI) seq(N) = {
%o my(a = vector(N)); a[1] = 0; a[2] = 1;
%o for (n = 3, N, a[n] = (2*n-1)*a[n-1] + a[n-2]);
%o concat(1, a);
%o };
%o seq(20) \\ _Gheorghe Coserea_, Dec 09 2016
%o (Magma) [n le 2 select 2-n else (2*n-3)*Self(n-1) + Self(n-2): n in [1..30]]; // _G. C. Greubel_, Sep 26 2023
%o (SageMath)
%o def A278990_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(-1+sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
%o A278990_list(30) # _G. C. Greubel_, Sep 26 2023
%Y Column k=0 of A079267.
%Y Column k=2 of A293157.
%Y Row n=2 of A322013.
%Y Cf. A000110, A000699 (topologically connected 2-uniform), A000806, A001147 (2-uniform), A003436 (cyclical version), A005493, A170941, A190823 (distance 3+ version), A322402, A324011, A324172.
%Y Anti-run compositions are A003242.
%Y Separable partitions are A325534.
%Y Cf. A007716, A292884, A333489.
%Y Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Dec 07 2016
%E a(0)=1 prepended by _Gheorghe Coserea_, Dec 09 2016
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