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 A133365 #19 Nov 26 2017 09:49:38
%S A133365 1,1,2,5,13,36,105,321,1018,3334,11216,38635,135835,486337,1769500,
%T A133365 6531796,24425758,92420026,353444218,1364933719,5318450239,
%U A133365 20894505025,82713826842,329746065427,1323179962753,5341963415921,21689519880470,88533441655211
%N A133365 Number of 3-noncrossing RNA structures, i.e., the number of 3-noncrossing partial matchings over n vertices and without arcs of length 1.
%C A133365 a(n) is the sum of entries in row n of the triangle in A187253.
%C A133365 a(n) is asymptotically equal to 4!*10.4724*((5+sqrt(21))/2)^n/(n(n-1)(n-2)(n-3)(n-4)).
%H A133365 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 A133365 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.
%H A133365 Emma Y. Jin and Christian M. Reidys, <a href="http://arxiv.org/abs/0706.3137">Asymptotic Enumeration of RNA Structures with Pseudoknots</a>, arXiv:0706.3137 [q-bio.BM], 2007.
%H A133365 Emma Y. Jin and Christian M. Reidys, <a href="http://dx.doi.org/10.1007/s11538-007-9265-2">Asymptotic Enumeration of RNA Structures with Pseudoknots</a>, Bulletin of Mathematical Biology 70 (2008), 951-970.
%H A133365 Emma Y. Jin and Christian M. Reidys, <a href="https://doi.org/10.1016/j.jtbi.2007.09.020">Central and local limit theorems for RNA structures</a>, J. Theoretical Biology, 250, 2008, 547-559.
%F A133365 a(n) = Sum_{k=0..n} T(n,k), where T(n,k) = Sum((-1)^j*binomial(n-j,j)*binomial(n-2j,k)*[c((n-k)/2-2j)*c((n-k)/2-j+2)-c((n-k)/2-j+1)^2], j=0..(n-k)/2), and c(n)=A000108(n) are the Catalan numbers. [Perhaps this formula is using the convention that c(x) = 0 unless x is a nonnegative integer? - _N. J. A. Sloane_, Jul 24 2017]
%e A133365 a(4)=5 because we have ABAB, AIAI, AIIA, IAIA, and IIII, where pairs of A's and pairs of B's are assumed to be joined by an arc and the I's are isolated vertices.
%p A133365 c := proc (n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T := proc (n, k) if `mod`(n-k, 2) = 0 then sum((-1)^b*binomial(n-b, b)*binomial(n-2*b, k)*(c((1/2)*n-(1/2)*k-b)*c((1/2)*n-(1/2)*k-b+2)-c((1/2)*n-(1/2)*k-b+1)^2), b = 0 .. (1/2)*n-(1/2)*k) else 0 end if end proc: seq(add(T(n, k), k = 0 .. n), n = 1 .. 28);
%t A133365 c = CatalanNumber;
%t A133365 T[n_, k_] := If[EvenQ[m = n-k], Sum[(-1)^b*Binomial[n-b, b] * Binomial[n - 2*b, k] * (c[m/2-b]*c[m/2-b+2] - c[m/2-b+1]^2), {b, 0, m/2}], 0];
%t A133365 a[n_] := Sum[T[n, k], {k, 0, n}];
%t A133365 Array[a, 28] (* _Jean-François Alcover_, Nov 26 2017, from Maple *)
%Y A133365 Cf. A000108, A187253.
%K A133365 nonn
%O A133365 1,3
%A A133365 Emma Y. Jin (emma(AT)cfc.nankai.edu.cn), Oct 26 2007