A278991 a(n) is the number of simple linear diagrams with n+1 chords.
0, 1, 3, 24, 211, 2325, 30198, 452809, 7695777, 146193678, 3069668575, 70595504859, 1764755571192, 47645601726541, 1381657584006399, 42829752879449400, 1413337528735664887, 49465522112961344241, 1830184115528550306438, 71375848864779552073957
Offset: 0
Keywords
Links
- Gheorghe Coserea, Table of n, a(n) for n = 0..301
- 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.
Programs
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Mathematica
a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := a[n] = (2 n - 1) a[n - 1] + (4 n - 3) a[n - 2] + (2 n - 4) a[n - 3]; Table[a@ n, {n, 0, 19}] (* Michael De Vlieger, Dec 10 2016 *) -
PARI
seq(N) = { my(a = vector(N)); a[1]=1; a[2]=3; a[3]=24; for (n=4, N, a[n] = (2*n-1)*a[n-1] + (4*n-3)*a[n-2] + (2*n-4)*a[n-3]); concat(0, a); }; seq(20) \\ Gheorghe Coserea, Dec 10 2016 -
PARI
N = 20; x = 'x + O('x^N); concat(0, Vec(serlaplace((1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x))))) \\ Gheorghe Coserea, Dec 10 2016
Formula
E.g.f.: (1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x)).
a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n+3/2). - Vaclav Kotesovec, Dec 07 2016
a(n) = (2*n-1)*a(n-1) + (4*n-3)*a(n-2) + (2*n-4)*a(n-3). - Gheorghe Coserea, Dec 10 2016
Extensions
Offset corrected by Gheorghe Coserea, Dec 10 2016