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 A187253 #11 Nov 11 2022 08:06:59
%S A187253 1,0,1,0,0,1,0,1,0,1,1,0,3,0,1,0,6,0,6,0,1,4,0,21,0,10,0,1,0,34,0,55,
%T A187253 0,15,0,1,22,0,157,0,120,0,21,0,1,0,232,0,526,0,231,0,28,0,1,139,0,
%U A187253 1317,0,1435,0,406,0,36,0,1,0,1761,0,5355,0,3388,0,666,0,45,0,1,979,0,11883,0,17500,0,7182,0,1035,0,55,0,1
%N A187253 Triangle read by rows: T(n,k) is the number of 3-noncrossing RNA structures on n vertices having k isolated vertices.
%C A187253 Sum of entries in row n is A133365(n).
%C A187253 T(n,k)=0 if n-k is odd.
%C A187253 T(n,0)=A187254(n).
%C A187253 Sum_{k=0..n} k*T(n,k) = A187255(n).
%H A187253 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 A187253 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.
%F A187253 T(n,k) = Sum_{j=0..(n-k)/2} (-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), where c(n)=A000108(n) are the Catalan numbers (see Corollary 2 in the Jin et al. reference).
%e A187253 T(4,2)=3 because we have AIAI, IAIA, AIIA, where in each structure the two A's are joined by an arc and the two I's are isolated vertices.
%e A187253 T(4,4)=1 because we have IIII.
%e A187253 T(4,0)=1 because we have ABAB, where the two A's are joined by an arc and the two B's are joined by an arc.
%e A187253 Triangle starts:
%e A187253 1;
%e A187253 0, 1;
%e A187253 0, 0, 1;
%e A187253 0, 1, 0, 1;
%e A187253 1, 0, 3, 0, 1;
%e A187253 0, 6, 0, 6, 0, 1;
%e A187253 4, 0, 21, 0, 10, 0, 1.
%p A187253 c := proc (n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: T := proc (n, l) if `mod`(n-l, 2) = 0 then sum((-1)^b*binomial(n-b, b)*binomial(n-2*b, l)*(c((1/2)*n-(1/2)*l-b)*c((1/2)*n-(1/2)*l-b+2)-c((1/2)*n-(1/2)*l-b+1)^2), b = 0 .. (1/2)*n-(1/2)*l) else 0 end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A187253 Cf. A000108, A133365, A187254, A187255.
%K A187253 nonn,tabl
%O A187253 0,13
%A A187253 _Emeric Deutsch_, Apr 24 2011