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%I A278991 #49 Nov 03 2021 06:03:33
%S A278991 0,1,3,24,211,2325,30198,452809,7695777,146193678,3069668575,
%T A278991 70595504859,1764755571192,47645601726541,1381657584006399,
%U A278991 42829752879449400,1413337528735664887,49465522112961344241,1830184115528550306438,71375848864779552073957
%N A278991 a(n) is the number of simple linear diagrams with n+1 chords.
%H A278991 Gheorghe Coserea, <a href="/A278991/b278991.txt">Table of n, a(n) for n = 0..301</a>
%H A278991 E. Krasko and A. Omelchenko, <a href="http://arxiv.org/abs/1601.05073">Enumeration of Chord Diagrams without Loops and Parallel Chords</a>, arXiv preprint arXiv:1601.05073 [math.CO], 2016.
%H A278991 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.
%F A278991 E.g.f.: (1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x)).
%F A278991 a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n+3/2). - _Vaclav Kotesovec_, Dec 07 2016
%F A278991 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
%t A278991 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 *)
%o A278991 (PARI)
%o A278991 seq(N) = {
%o A278991   my(a = vector(N)); a[1]=1; a[2]=3; a[3]=24;
%o A278991   for (n=4, N, a[n] = (2*n-1)*a[n-1] + (4*n-3)*a[n-2] + (2*n-4)*a[n-3]);
%o A278991   concat(0, a);
%o A278991 };
%o A278991 seq(20) \\ _Gheorghe Coserea_, Dec 10 2016
%o A278991 (PARI)
%o A278991 N = 20; x = 'x + O('x^N);
%o A278991 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
%Y A278991 Cf. A003436, A003437, A007474, A278990, A278992, A278993, A278994.
%K A278991 nonn
%O A278991 0,3
%A A278991 _N. J. A. Sloane_, Dec 07 2016
%E A278991 Offset corrected by _Gheorghe Coserea_, Dec 10 2016

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