A278992 Number of simple chord-labeled chord diagrams with n chords.

0, 1, 1, 21, 168, 1968, 26094, 398653, 6872377, 132050271, 2798695656, 64866063276, 1632224748984, 44316286165297, 1291392786926821, 40202651019430461, 1331640833909877144, 46762037794122159492, 1735328399106396110310, 67858430028772637693845

Offset

1,4

Links

Table of n, a(n) for n=1..20.

E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv preprint 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.

Formula

E.g.f.: (1+sqrt(1-2*t))*(1-2*t)^(-1/2)*exp(-1-t+sqrt(1-2*t))-(2-t)*exp(-t).

a(n) ~ 2^(n+1/2) * n^n / exp(n+3/2). - Vaclav Kotesovec, Dec 07 2016

Conjecture D-finite with recurrence: +(-n+2)*a(n) +(2*n^2-8*n+7)*a(n-1) +(6*n^2-18*n+11)*a(n-2) +(n-1)*(6*n-11)*a(n-3) +2*(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jan 27 2020

Mathematica

terms = 20;
CoefficientList[(Sqrt[1 - 2t]+1)(1/Sqrt[1 - 2t])*E^(Sqrt[1 - 2t] - t - 1) - (2-t)/E^t + O[t]^(terms+1), t]*Range[0, terms]! // Rest (* Jean-François Alcover, Sep 14 2018 *)

Crossrefs

Cf. A003436, A003437, A007474, A278990, A278991, A278993, A278994.

Sequence in context: A266733 A107970 A105249 * A358930 A041848 A125358

Adjacent sequences: A278989 A278990 A278991 * A278993 A278994 A278995

Keyword

nonn

Author

N. J. A. Sloane, Dec 07 2016

Status

approved