A278990 Number of loopless linear chord diagrams with n chords.

1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096, 4939227215, 113836841041, 2850860253240, 77087063678521, 2238375706930349, 69466733978519340, 2294640596998068569, 80381887628910919255, 2976424482866702081004, 116160936719430292078411

Offset: 0

Author

N. J. A. Sloane, Dec 07 2016

Keywords

• nonn

Comments

See the signed version of these numbers, A000806, for much more information about these numbers.

From Gus Wiseman, Feb 27 2019: (Start)

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:

{{1,3},{2,5},{4,6}}

{{1,4},{2,5},{3,6}}

{{1,4},{2,6},{3,5}}

{{1,5},{2,4},{3,6}}

{{1,6},{2,4},{3,5}}

(End)

From Gus Wiseman, Jul 05 2020: (Start)

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.

(1,2,3,1,2,3)

(1,2,3,1,3,2)

(1,2,3,2,1,3)

(1,2,3,2,3,1)

(1,2,1,3,2,3)

(End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..404 (terms 0..200 from Gheorghe Coserea)
• Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
• H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
• E. Krasko, I. Labutin, and A. Omelchenko, Enumeration of labelled and unlabelled Hamiltonian Cycles in complete k-partite graphs, arXiv:1709.03218 [math.CO], 2017, Table 1.
• E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv:1601.05073 [math.CO], 2016.
• E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
• Gus Wiseman, The a(4) = 36 loopless linear chord diagrams.
• Donovan Young, Counting Bubbles in Linear Chord Diagrams, arXiv:2311.01569 [math.CO], 2023.
• Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

Crossrefs

Column k=0 of A079267.

Column k=2 of A293157.

Row n=2 of A322013.

Cf. A000110, A000699 (topologically connected 2-uniform), A000806, A001147 (2-uniform), A003436 (cyclical version), A005493, A170941, A190823 (distance 3+ version), A322402, A324011, A324172.

Anti-run compositions are A003242.

Separable partitions are A325534.

Cf. A007716, A292884, A333489.

Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.

Programs

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

Mathematica

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 *)
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 *)
Table[(2 n-1)!! Hypergeometric1F1[-n,-2 n,-2], {n,0,20}] (* Eric W. Weisstein, Nov 14 2018 *)
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 *)
twouniflin[{}]:={{}};twouniflin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@twouniflin[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+1&]}];
Table[Length[twouniflin[Range[n]]],{n,0,14,2}] (* Gus Wiseman, Feb 27 2019 *)

PARI

seq(N) = {
  my(a = vector(N)); a[1] = 0; a[2] = 1;
  for (n = 3, N, a[n] = (2*n-1)*a[n-1] + a[n-2]);
  concat(1, a);
};
seq(20) \\ Gheorghe Coserea, Dec 09 2016

SageMath

def A278990_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-1+sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
A278990_list(30) # G. C. Greubel, Sep 26 2023

Formula

From Gheorghe Coserea, Dec 09 2016: (Start)

D-finite with recurrence a(n) = (2*n-1)*a(n-1) + a(n-2), with a(0) = 1, a(1) = 0.

E.g.f. y satisfies: 0 = (1-2*x)*y'' - 3*y' - y.

a(n) - a(n-1) = A003436(n) for all n >= 2. (End)

From Vaclav Kotesovec, Sep 15 2017: (Start)

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.

a(n) ~ 2^(n+1/2) * n^n / exp(n+1). (End)

a(n) = A114938(n)/n! - Gus Wiseman, Jul 05 2020 (from Alexander Burstein's formula at A114938).

From G. C. Greubel, Sep 26 2023: (Start)

a(n) = (-1)^n * (i/e)*Sqrt(2/Pi) * BesselK(n + 1/2, -1).

G.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * Erfi((1+x)/sqrt(2*x)).

E.g.f.: exp(-1 + sqrt(1-2*x))/sqrt(1-2*x). (End)

Extensions

a(0)=1 prepended by Gheorghe Coserea, Dec 09 2016