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A278990 Number of loopless linear chord diagrams with n chords. 13
1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096, 4939227215, 113836841041, 2850860253240, 77087063678521, 2238375706930349, 69466733978519340, 2294640596998068569, 80381887628910919255, 2976424482866702081004, 116160936719430292078411 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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)

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, A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.

E. Krasko, I. Labutin, A. Omelchenko, Enumeration of labelled and unlabelled Hamiltonian Cycles in complete k-partite graphs, arXiv:1709.03218, 2017, Table 1.

E. Krasko, A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv:1601.05073 [math.CO], 2016.

E. Krasko, 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.

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)

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, 18}]] (* 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 *)

PROG

(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

CROSSREFS

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.

Sequence in context: A300987 A067305 A000806 * A127132 A141764 A075744

Adjacent sequences:  A278987 A278988 A278989 * A278991 A278992 A278993

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane, Dec 07 2016

EXTENSIONS

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

STATUS

approved

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Last modified February 23 06:50 EST 2020. Contains 332159 sequences. (Running on oeis4.)