This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103140 #15 Nov 12 2023 00:06:34
%S A103140 1,1,1,2,5,14,40,119,364,1145,3688,12139,40734,139071,482214,1695469,
%T A103140 6036768,21740969,79117822,290674470,1077306351,4025068621,
%U A103140 15151115808,57427176992,219068962330,840708048210,3244438898552,12586627632549,49069788882951
%N A103140 Number of 3-noncrossing restricted RNA structures with n vertices.
%C A103140 a(n) is the number of 3-noncrossing partial matchings over n vertices and without arcs of length 1 and 2. - _Andrey Zabolotskiy_, Nov 11 2023
%H A103140 Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/patterns2019.pdf">Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata</a>, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
%H A103140 Emma Y. Jin, Jing Qin and Christian M. Reidys, <a href="https://arxiv.org/abs/0704.2518">Combinatorics of RNA structures with pseudoknots</a>, arXiv:0704.2518 [math.CO], 2007.
%H A103140 Emma Y. Jin, Jing Qin and Christian M. Reidys, <a href="https://doi.org/10.1007/s11538-007-9240-y">Combinatorics of RNA structures with pseudoknots</a>, Bulletin of Mathematical Biology Vol. 70 (2008) pp. 45-67. See Table 2 on page 62 for details.
%t A103140 sf3[n_] := sf3[n] = Sum[Binomial[n, 2 k] (CatalanNumber[k + 2] CatalanNumber[k] - CatalanNumber[k + 1]^2), {k, 0, n/2}]; (* this is A049401 *)
%t A103140 la[0, 0, 0] = 1;
%t A103140 la[_?Negative, _, _] = la[_, _?Negative, _] = la[_, _, _?Negative] = 0;
%t A103140 la[n_, b1_, b2_] := la[n, b1, b2] = la[n - 2, b1 - 1, b2] + la[n - 1, b1, b2] + la[n - 4, b1, b2 - 2] + la[n - 3, b1, b2 - 1];
%t A103140 a[n_] := Sum[(-1)^(b1 + b2) la[n, b1, b2] sf3[n - 2 (b1 + b2)], {b1, 0, n/2}, {b2, 0, n/2}];
%t A103140 Table[a[n], {n, 30}] (* _Andrey Zabolotskiy_, Nov 11 2023, from eqs. (4.2), (4.3), and (2.14) by Jin et al. *)
%Y A103140 Cf. A133365, A049401.
%K A103140 nonn
%O A103140 1,4
%A A103140 _Parthasarathy Nambi_, Sep 07 2008
%E A103140 Terms a(16) and beyond from _Andrey Zabolotskiy_, Nov 11 2023